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A concise guide to complex Hadamard matrices

Wojciech Tadej1 and Karol Zyczkowski2,3

1Faculty of Mathematics and Natural Sciences, College of Sciences,

Cardinal Stefan Wyszynski University, Warsaw, Poland

2Institute of Physics, Jagiellonian University, Krakow, Poland

3Center for Theoretical Physics, Polish Academy of Sciences, Warsaw, Poland

e-mail: wtadej@wp.pl karol@tatry.if.uj.edu.pl

January 26, 2006

Abstract

Complex Hadamard matrices, consisting of unimodular entries with

arbitrary phases, play an important role in the theory of quantum in-

formation. We review basic properties of complex Hadamard matrices

and present a catalogue of inequivalent cases known for dimension N =

2, . . . , 16. In particular, we explicitly write down some families of complex

Hadamard matrices for N = 12, 14 and 16, which we could not find in the

existing literature.

1 Introduction

In a 1867 paper on ’simultaneous sign-successions, tessellated pavements in twoor more colors, and ornamental tile-work’ Sylvester used self–reciprocial matri-ces, defined as a square array of elements of which each is proportional to its firstminor [1]. This wide class of matrices includes in particular these with orthogo-nal rows and columns. In 1893 Hadamard proved that such matrices attain thelargest value of the determinant among all matrices with entries bounded byunity [2]. After this paper the matrices with entries equal to ±1 and mutuallyorthogonal rows and columns were called Hadamard.

Originally there was an interest in real Hadamard matrices, Hij ∈ R, whichfound diverse mathematical applications, in particular in error correction andcoding theory [3] and in the design of statistical experiments [4]. Hadamardproved that such matrices may exist only for N = 2 or for size N being amultiple of 4 and conjectured that they exist for all such N . A huge collectionof real Hadamard matrices for small values of N is known (see e.g. [5, 6]), butthe original Hadamard conjecture remains unproven. After a recent discovery

1

of N = 428 real Hadamard matrix [7], the case N = 668 is the smallest order,for which the existence problem remains open.

Real Hadamard matrices may be generalized in various ways. Butson in-troduced a set H(q, N) of Hadamard matrices of order N , the entries of whichare q-th roots of unity [8, 9]. Thus H(2, N) represents real Hadamard matrices,while H(4, N) denotes Hadamard matrices1 with entries ±1 or ±i. If p is prime,then H(p, N) can exist if N = mp with an integer m [8], and it is conjecturedthat they exist for all such cases [14].

In the simplest case m = 1 Hadamard matrices H(N, N) exist for any di-mension. An explicit construction based on Fourier matrices

[F ′N ]j,k :=

1√N

ei(j−1)(k−1) 2πN with j, k ∈ {1, . . . , N} (1)

works for an arbitrary N , not necessarily prime. A Fourier matrix is unitary,but a rescaled matrix FN =

√NF ′

N belongs to H(N, N). In the following we aregoing to use Hadamard matrices with unimodular entries, but for conveniencewe shall also refer to FN as Fourier matrix.

For m = 2 it is not difficult [8] to construct matrices H(p, 2p). In general,the problem of finding all pairs {q, N} for which Butson–type matrices H(q, N)do exist, remains unsolved [15], even though some results on non-existence areavailable [16]. The set of p–th roots of unity forms a finite group and it ispossible to generalize the notion of Butson–type matrices for any finite group[17, 18, 19].

In this work we will be interested in a more general case of complexHadamard matrices, for which there are no restrictions on phases of eachentry of H . Such a matrix, also called biunitary [20], corresponds to takingthe limit q → ∞ in the definition of Butson. In this case there is no analogueof Hadamard conjecture, since complex Hadamard matrices exists for any di-mension. However, in spite of many years of research, the problem of findingall complex Hadamard matrices of a given size N is still open [21, 22].

Note that this problem may be considered as a particular case of a moregeneral issue of specifying all unitary matrices such that their squared moduligive a fixed doubly stochastic matrix [23, 24]. This very problem was intensivelystudied by high energy physicists investigating the parity violation and analyzingthe Cabibbo–Kobayashi–Maskawa matrices [25, 26, 27, 28].

On one hand, the search for complex Hadamard matrices is closely related tovarious mathematical problems, including construction of some ∗-subalgebras infinite von Neumann algebras [29, 30, 31, 21], analyzing bi-unimodular sequencesor finding cyclic n–roots [32, 33] and equiangular lines [34]. Complex Hadamardmatrices were used to construct error correcting codes [15] and to investigatethe spectral sets and Fuglede’s conjecture [35, 36, 37, 38].

On the other hand, Hadamard matrices find numerous applications in severalproblems of theoretical physics. For instance, Hadamard matrices (rescaled

1These matrices were also called complex Hadamard matrices [10, 11, 12, 13].

2

by 1/√

N to achieve unitarity), are known in quantum optics as symmetricmultiports [40, 41] (and are sometimes called Zeilinger matrices) and may beused to construct spin models [34], or schemes of selective coupling of a multi–qubit system [42].

Complex Hadamard matrices play a crucial role in the theory of quantum

information as shown in a seminal paper of Werner [43]. They are used insolving the Mean King Problem [44, 45, 46], and in finding ’quantum designs’[47]. Furthermore, they allow one to construct

a) Bases of unitary operators, i.e. the set of mutually orthogonal

unitary operators, {Uk}N2

k=1 such that Uk ∈ U(N) and TrU †kUl = Nδkl for

k, l = 1, . . . , N2,b) Bases of maximally entangled states, i.e. the set {|Ψk〉}N2

k=1 suchthat each |Ψk〉 belongs to a composed Hilbert space with the partial traceTrN(|Ψk〉〈Ψk|) = 1/N , and they are mutually orthogonal, 〈Ψk|Ψl〉 = δkl [48],

c) Unitary depolarisers, i.e. the set {Uk}N2

k=1 such that for any bounded

linear operator A the property∑N2

k=1 U †kAUk = NTrA 1 holds.

The problems a)-c) are equivalent in the sense that given a solution to oneproblem one can find a solution to the other one, as well as a correspondingscheme of teleportation or dense coding [43]. In particular Hadamard matricesare usefull to construct a special class of unitary bases of a group type, alsocalled ’nice error basis’ [49, 50]

Another application of Hadamard matrices is related to quantum tomog-raphy: To determine all N2 − 1 parameters characterizing a density matrixof size N one needs to perform k ≥ N + 1 orthogonal measurements. Each

measurement can be specified by an orthogonal basis Φu = {|φ(u)i 〉}N

i=1 set foru = 1, . . . , k. Precision of such a measurement scheme is optimal if the basesare mutually unbiased, i.e. they are such that

|〈φ(u)i |φ(s)

j 〉|2 =1

N(1 − δus) + δusδij . (2)

If the dimension N is prime or a power of prime the number of maximallyunbiased bases (MUB) is equal to N + 1 [51, 52], but for other dimensionsthe answer to this question is still unknown [53, 54, 55]. The task of finding(k + 1) MUBs is equivalent to finding a collection of k mutually unbiasedHadamards (MUH),

{Hi ∈ HN}ki=1 :

1√N

H†i Hj ∈ HN , i > j = 1, . . . , k − 1, (3)

since the set {1, H1/√

N, . . . , Hk/√

N} forms a set of MUBs. Here HN denotesthe set of complex2 Hadamard matrices of size N .

The aim of this work is to review properties of complex Hadamard matricesand to provide a handy collection of these matrices of size ranging from 2 to

2Similarly, knowing unbiased real Hadamard matrices one constructs real MUBs [56].

3

16. Not only we list concrete Hadamard matrices, the existence of which followsfrom recent papers by Haagerup [21] and Dita [22], but also we present severalother Hadamard matrices which have not appeared in the literature so far.

2 Equivalent Hadamard matrices and the de-

phased form

We shall start this section providing some formal definitions.

Definition 2.1 A square matrix H of size N consisting of unimodular entries,|Hij | = 1, is called a Hadamard matrix if

HH† = N 1 , (4)

where † denotes the Hermitian transpose. One distinguishes

a) real Hadamard matrices, Hij ∈ R, for i, j = 1, . . . , N ,

b) Hadamard matrices of Butson type H(q, N), for which (Hij)q = 1,

c) complex Hadamard matrices, Hij ∈ C.

The set of all Hadamard matrices of size N will be denoted by HN .

Definition 2.2 Two Hadamard matrices H1 and H2 are called equivalent,written H1 ≃ H2, if there exist diagonal unitary matrices D1 and D2 and per-mutations matrices P1 and P2 such that [21]

H1 = D1P1 H2 P2D2 . (5)

This equivalence relation may be considered as a generalization of the Hadamardequivalence in the set of real Hadamard matrices, in which permutations andnegations of rows and columns are allowed.3

Definition 2.3 A complex Hadamard matrix is called dephased when the en-tries of its first row and column are all equal to unity4,

H1,i = Hi,1 = 1 for i = 1, . . . , N . (6)

Remark 2.4 For any complex N ×N Hadamard matrix H there exist uniquelydetermined diagonal unitary matrices, Dr = diag(H11, H21, . . . , HN1), and Dc =diag(1, H11H12, . . . , H11H1N ), such that [Dc]1,1 = 1 and

Dr · H · Dc (7)

is dephased.3Such an equivalence relation may be extended to include also transposition and complex

conjugation [28]. Since the transposition of a matrix is not realizable in physical systems weprefer to stick to the original definition of equivalence.

4in case of real Hadamard matrices such a form is called normalised.

4

Two Hadamard matrices with the same dephased form are equivalent. Thusthe relevant information on a Hadamard matrix is carried by the lower rightsubmatrix of size N − 1, called the core [9].

It is often useful to define a log–Hadamard matrix Φ, such that

Hkl = eiΦkl , (8)

is Hadamard. The phases Φkl entering a log-Hadamard matrix may be chosento belong to [0, 2π). This choice of phases implies that the matrix qΦ/2π cor-responding to a Hadamard matrix of the Butson type H(q, N) consists of zerosand integers smaller than q. All the entries of the first row and column of a logHadamard matrix Φ, corresponding to a dephased Hadamard matrix, are equalto zero.

To illustrate the procedure of dephasing consider the Fourier-like matrix

of size four,[

F4

]

j,k:= eijk2π/4 where j, k ∈ {1, 2, 3, 4}. Due to this choice

of entries the matrix F4 is not dephased, but after operation (7) it takes thedephased form F4,

F4 =

i −1 −i 1−1 1 −1 1−i −1 i 1

1 1 1 1

, F4 =

1 1 1 11 i −1 −i1 −1 1 −11 −i −1 i

. (9)

The corresponding log–Hadamard matrices read

Φ4 =2π

4

1 2 3 02 0 2 03 2 1 00 0 0 0

, Φ4 =2π

4

0 0 0 00 1 2 30 2 0 20 3 2 1

. (10)

Note that in this case dephasing is equivalent to certain permutation of rows andcolumns, but in general the role of both operations is different. It is straight-forward to perform (7) which brings any Hadamard matrix into the dephasedform. However, having two matrices in such a form it might not be easy toverify, whether there exist permutations P1 and P2 necessary to establish theequivalence relation (5). This problem becomes difficult for larger matrix sizes,since the number of possible permutations grows as N !. What is more, com-bined multiplication by unitary diagonal and permutation matrices may still benecessary.

To find necessary conditions for equivalence one may introduce various in-variants of operations allowed in (5), and compare them for two matrices in-vestigated. In the case of real Hadamard matrices and permutations only, thenumber of negative elements in the dephased form may serve for this purpose.Some more advanced methods for detecting inequivalence were recently pro-posed by Fang and Ge [57]. It has been known for many years that for N = 4, 8and 12 all real Hadamard matrices are equivalent, while the number of equiva-lence classes for N = 16, 20, 24 and 28 is equal to 5, 3, 60 and 487, respectively.

5

For higher dimensions this number grows dramatically: For N = 32 and 36 thenumber of inequivalent matrices is not smaller than 3, 578, 006 and 4, 745, 357,but the problem of enumerating all of them remains open – see e.g. [58].

To characterize a complex Hadamard matrix H let us define a set of coeffi-cients,

Λ = {Λµ := HijHkjHklHil : µ = (i, j, k, l) ∈ {1, . . . , N}×4} , (11)

where no summation over repeating indeces is assumed and µ stands for a com-posite index (i, j, k, l). Due to complex conjugation of the even factors in theabove definition any concrete value of Λµ is invariant with respect to multipli-cation by diagonal unitary matrices.5 Although this value may be altered by apermutation, the entire set Λ is invariant with respect to operations allowed in(5). This fact allows us to state the Haagerup condition for inequivalence [21],

Lemma 2.5 If two Hadamard matrices have different sets Λ of invariants (11),they are not equivalent.

The above criterion works in one direction only. For example, any Hadamardmatrix H and its transpose, HT , possess the same sets Λ, but they need not tobe permutation equivalent.

Some other equivalence criteria are dedicated to certain special cases. Thetensor product of two Hadamard matrices is also a Hadamard matrix:

H1 ∈ HM and H2 ∈ HN =⇒ H1 ⊗ H2 ∈ HMN (12)

and this fact will be used to construct Hadamard matrices of a larger size. Ofparticular importance are tensor products of Fourier matrices, FM⊗FN ∈ HMN .For arbitrary dimensions both tensor products are equivalent, FN ⊗FM ≃ FM ⊗FN . However, their equivalence with FMN depends on the number theoreticproperty of the product M ·N : the equivalence holds if M and N are relativelyprime [59].

To classify and compare various Hadamard matrices we are going to usethe dephased form (6) thus fixing the first row and column in each comparedmatrix. However, the freedom of permutation inside the remaining submatrixof size (N − 1) does not allow us to specify a unique ’canonical form’ for agiven complex Hadamard matrix. In other words we are going to list onlycertain representatives of each equivalence class known, but the reader has tobe warned that other choices of equivalent representatives are equally legitimate.For instance, the one parameter family of N = 4 complex Hadamard matricespresented in [21, 22] contains all parameter-dependent elements of H in its lower-right corner, while our choice (65) with variable phases in second and fourthrow is due to the fact that such an orbit stems from the Fourier matrix F4.

Identifying matrices equivalent with respect to (5) we denote by

GN = HN/ ≃ (13)5Such invariants were used by physicists investigating unitary Kobayashi–Maskawa matri-

ces of size 3 and 4 [25, 26, 27, 28].

6

every set of representatives of different equivalence classes.

Interestingly GN is known only for N = 2, 3, 4, 5, while the problem of findingall complex Hadamard matrices for N ≥ 6 remains unsolved. In particular,compiling our list of Hadamard matrices we took into account all continuousfamilies known to us, but in several cases it is not clear, whether there exist anyequivalence relations between them.

3 Isolated Hadamard matrices and continuous

orbits of inequivalent matrices

In this and the following sections we shall use the symbol ◦ to denote theHadamard product of two matrices,

[H1 ◦ H2]i,j = [H1]i,j · [H2]i,j , (14)

and the EXP symbol to denote the entrywise exp operation on a matrix,

[EXP(R)]i,j = exp(

[R]i,j

)

. (15)

3.1 Isolated Hadamard matrices

Definition 3.1 A dephased N ×N complex Hadamard matrix H is called iso-lated if there is a neighbourhood W around H such that there are no otherdephased complex Hadamard matrices in W.

To have a tool useful in determining whether a given dephased complexHadamard matrix of size N is isolated, we introduce the notion of defect:

Definition 3.2 The defect d(H) of an N × N complex Hadamard matrix His the dimension of the solution space of the real linear system with respect to amatrix variable R ∈ RN2

:

R1,j = 0 j ∈ {2, . . . , N} (16)

Ri,1 = 0 i ∈ {1, . . . , N} (17)

N∑

k=1

Hi,kHj,k (Ri,k − Rj,k) = 0 1 ≤ i < j ≤ N (18)

The defect allows us to formulate a one way criterion:

Lemma 3.3 A dephased complex Hadamard matrix H is isolated if the defectof H is equal to zero.

If N is prime then the defect of FN is zero, so the Fourier matrix is isolated,as earlier shown in [20, 62]. For any composed N the defect of the Fouriermatrix is positive. For instance, if the dimension N is a product of two distinct

7

primes, then d(Fpq) = 2(p− 1)(q− 1), while for powers of a prime, N = pk withk ≥ 2, the defect reads

d(Fpk ) = pk−1[k(p − 1) − p] + 1 . (19)

An explicit formula for d(FN ) for an arbitrary composed N is derived elsewhere[63]. In that case the Fourier matrix belongs to a continuous family, as examplesshow in section 5.

The reasoning behind the criterion of Lemma 3.3 runs as follows:

Any dephased complex Hadamard matrices, in particular those in a neigh-bourhood of a dephased complex Hadamard matrix H , must be of the form:

H ◦EXP(i · R) (20)

where an n× n real matrix R satisfies ’dephased property’ and unitarity condi-tions for (20):

R1,j = 0 j ∈ {2, . . . , N} (21)

Ri,1 = 0 i ∈ {1, . . . , N} (22)

−i ·N∑

k=1

Hi,kHj,kei(Ri,k−Rj,k) = 0 1 ≤ i < j ≤ N (23)

We will rewrite these conditions using a real vector function f , whose co-ordinate functions will be indexed by the values (symbol sequences) from theset I, related to the standard index set by a fixed bijection (one to one map)β : I −→ {1, 2, . . . , (2N − 1) + (N2 − N)}. The set I reads:

I = { (1, 2), (1, 3), . . . , (1, N) } ∪ { (1, 1), (2, 1), . . . , (N, 1) }∪ { (i, j, t) : 1 ≤ i < j ≤ N and t ∈ {Re, Im} } (24)

For simplicity of notation we further write fi, i ∈ I to denote fβ(i).Similarly, a fixed bijection α : {1, . . . , N} × {1, . . . , N} −→ {1, . . . , N2}

allows matrix indexing the components of a real N2 element vector variable R,an argument to f , and we write Rk,l to denote Rα(k,l)

The (2N−1)+(N2−N) element function vector f is defined by the formulas:

f(1,j) = R1,j j ∈ {2, . . . , N} (25)

f(i,1) = Ri,1 i ∈ {1, . . . , N} (26)

f(i,j,Re) = Re

(

−i ·N∑

k=1

Hi,kHj,kei(Ri,k−Rj,k)

)

1 ≤ i < j ≤ N (27)

f(i,j,Im) = Im

(

−i ·N∑

k=1

Hi,kHj,kei(Ri,k−Rj,k)

)

1 ≤ i < j ≤ N (28)

8

Conditions (21,22,23) can now be rewritten as:

f(R) = 0 (29)

where the α(i, j)-th coordinate of a real variable vector R represents the i, j-thentry of the corresponding matrix R sitting in (20).

The value of the linear map Df0

: RN2 −→ R(2N−1)+(N2−1), being thedifferential of f at 0, at R is the vector:

[Df0(R)](1,j) = R1,j j ∈ {2, . . . , N} (30)

[Df0(R)](i,1) = Ri,1 i ∈ {1, . . . , N} (31)

[Df0(R)](i,j,Re) = Re

(

N∑

k=1

Hi,kHj,k (Ri,k − Rj,k)

)

1 ≤ i < j ≤ N (32)

[Df0(R)](i,j,Im) = Im

(

N∑

k=1

Hi,kHj,k (Ri,k − Rj,k)

)

1 ≤ i < j ≤ N (33)

where again indexing for f defined by β is used.

It is clear now that the kernel of the differential, {R ∈ RN2

: Df0(R) = 0},

corresponds to the solution space of system (16,17,18), in which R now takesthe meaning of an input variable vector to f , with indexing determined by α

Note that the N2 − N equation subsystem (18):

N∑

k=1

Hi,kHj,k (Ri,k − Rj,k) = 0 1 ≤ i < j ≤ N (34)

is solved at least by the (2N − 1)-dimensional real space spanned by 2N − 1vectors, defined by:

Rk,l =

{

1 for (k, l) ∈ {1, . . . , N} × {j}0 otherwise

, j ∈ {2, . . . , N} (35)

and

Rk,l =

{

1 for (k, l) ∈ {i} × {1, . . . , N}0 otherwise

, i ∈ {1, . . . , N} (36)

that is, by vectors, if treated as matrices (with the i, j-th entry being equal tothe α(i, j)-th coordinate of the corresponding variable vector), forming matriceswith either a row or a column filled all with 1’s and the other entries being 0’s.

If the defect of H equals 0, then the overall system (16,17,18) is solved only

by 0, the differential Df0

has full rank N2, i.e. dim(

Df0(RN2

))

= N2, and we

can choose an N2 equation subsystem:

f(R) = 0 (37)

9

of (29) such that the differential Df0

at 0 is of rank N2, i.e. dim(

Df0(RN2

))

=

N2, and thus f satisfies The Inverse Function Theorem. The theorem impliesin our case that, in a neighbourhood of 0, R = 0 is the only solution to (37),as well as the only solution to (29).

Also in this case, let us consider the differential, at 0, of the partial functionvector fU , given by (27,28). This differential value at R, DfU

0(R), is given

by the partial vector (32,33). Since the ’remaining’ differential, correspondingto the dephased ’property condition’ part of f , defined by (30,31), is of rank(2N−1), the rank of DfU

0 is equal to N2−(2N−1). Recall that from consideringabove the minimal solution space of system (18), it cannot be greater thanN2 − (2N − 1). Were it smaller, the rank of Df

0would be smaller than N2,

which cannot be if the defect of H is 0, see above.Then one can choose an N2 − (2N − 1) equation subsystem fU(R) = 0

of system fU(R) = 0, with the full rank:

dim(

DfU0(RN2

))

= N2 − (2N − 1) (38)

thus defining a (2N − 1) dimensional manifold around 0. This manifold gener-ates, by (20), the (2N − 1) dimensional manifold containing all, not necessarilydephased, complex Hadamard matrices in a neighbourhood of H . In fact, thelatter manifold is equal, around H , to the (2N − 1) dimensional manifold ofmatrices obtained by left and right multiplication of H by unitary diagonalmatrices:

{

diag(eiα1 , . . . , eiαN ) · H · diag(1, eiβ2 , . . . , eiβN )}

(39)

3.2 Continuous orbits of Hadamard matrices

The set of inequivalent Hadamard matrices is finite for N = 2 and N = 3, butalready for N = 4 there exists a continuous, one parameter family of equivalenceclasses. To characterize such orbits we will introduce the notion of an affineHadamard family.

Definition 3.4 An affine Hadamard family H(R) stemming from a de-phased N × N complex Hadamard matrix H is the set of matrices satisfying(4), associated with a subspace R of a space of all real N × N matrices withzeros in the first row and column,

H(R) = {H ◦ EXP(i · R) : R ∈ R} . (40)

The words ’family’ and ’orbit’ denote submanifolds of R2n2

consisting purelyof dephased complex Hadamard matrices. We will often write H(α1, . . . , αm) if

10

R is known to be an m-dimensional space with basis R1, . . . , Rm. In this case,H(α1, . . . , αm) will also denote the element of an affine Hadamard family:

H(α1, . . . , αm) = H(R)def= H ◦ EXP(i · R) (41)

where R = α1 · R1 + . . . + αm · Rm

An affine Hadamard family H(R) stemming from a dephased N×N complexHadamard matrix H is called a maximal affine Hadamard family when itis not contained in any larger affine Hadamard family H(R′) stemming from H :

R ⊂ R′ =⇒ R = R′ (42)

Calculation of the defect of H , defined in the previous section, is a steptowards determination of affine Hadamard families stemming from H :

Lemma 3.5 There are no affine Hadamard families stemming from a dephasedN × N complex Hadamard matrix H if it is isolated, in particular if the defectof H is equal to 0.

Lemma 3.6 The dimension of a continuous Hadamard orbit stemming from adephased Hadamard matrix H is not greater than the defect, d(H).

A lower bound dc(N) for the maximal dimensionality of a continuous orbitof inequivalent Hadamard matrices of size N was derived by Dita [22]. Inter-estingly, for powers of a prime, N = pk, this bound coincides with the defect(19) calculated at the Fourier matrix, which provides an upper bound for thedimension of an orbit stemming from FN . Thus in this very case the problemof determining dc(N) restricted to orbits including FN is solved and we knowthat the maximal affine Hadamard family stemming from FN is not containedin any non-affine orbit of a larger dimension.

Finally, we introduce two notions of equivalence between affine Hadamardfamilies:

Definition 3.7 Two affine Hadamard families stemming from dephased N ×Ncomplex Hadamard matrices H1 and H2: H1(R′) and H2(R′′), associated withreal matrix spaces R′ and R′′ of the same dimension, are called permutationequivalent if

there exist two permutation matrices Pr and Pc such that

H2(R′′) = Pr · H1(R′) · Pc (43)

i.e. there is one to one correspondence, by row and column permutation, betweenthe elements of H1(R′) and H2(R′′).

11

Note that permutation matrices considered in the above definition must notshift the first row or column of a matrix.

Definition 3.8 Two affine Hadamard families H1(R′) and H2(R′′) stemmingfrom dephased N ×N complex Hadamard matrices H1 and H2, associated withreal matrix spaces R′ and R′′ of the same dimension, are called cognate if

∀B ∈ H2(R′′) ∃A ∈ H1(R′) B ≃ AT , (44)

∀A ∈ H1(R′) ∃B ∈ H2(R′′) A ≃ BT . (45)

The family H(R) is called self–cognate if

∀B ∈ H(R) ∃A ∈ H(R) B ≃ AT (46)

4 Construction of Hadamard matrices

4.1 Same matrix size: reordering of entries and conjuga-

tion

If H is a dephased Hadamard matrix, so are its transpose HT , the conjugatedmatrix H and Hermitian transpose H†. It is not at all obvious, whether anypair of these is an equivalent pair. However, in some special cases it is so.

For example, the dephased forms of a Hadamard circulant matrix C and itstranspose CT are equivalent since C and CT are always permutation equivalent(see also the remark on the top of p.319 in [21]):

CT = PT · C · P (47)

where C is an N ×N circulant matrix Ci,j = xi−j mod N for some x ∈ CN , andP = [e1, eN , eN−1, . . . , e2], where ei are the standard basis column vectors.

On the other hand, there are infinitely many examples of equivalent andinequivalent pairs of circulant Hadamard matrices C, C, the same applying totheir dephased forms.

Apart from transposition and conjugation, for certain dimensions there existother matrix reorderings that preserve the Hadamard structure. Such operationsthat switch substructures of real Hadamard matrices to generate inequivalentmatrices have recently been discussed in [58]. It is likely that these methodsmay be useful to get inequivalent complex Hadamard matrices. In this way onlya finite number of inequivalent matrices can be obtained.

4.2 Same matrix size: linear variation of phases

Starting from a given Hadamard matrix H in the dephased form one may inves-tigate, whether it is possible to perform infinitesimal changes of some of (N−1)2

phases of the core of H to preserve unitarity. Assuming that all these phases(Φkl, from Eq. (8) k, l = 2, . . . , N) vary linearly with free parameters one can

12

find analytical form of such orbits, i.e. affine Hadamard families, stemming frome.g. Fourier matrices of composite dimensions [60].

To obtain affine Hadamard families stemming from H , one has to considerall pairs of rows of H . Now, taking the inner product of the rows in the i, j-thpair (1 ≤ i < j ≤ N), one gets zero as the sum of the corresponding values inthe sequence:

(

Hi,1 · Hj,1 , Hi,2 · Hj,2 , . . . , Hi,N · Hj,N

)

(48)

Such sequences will further be called chains, their subsequences – subchains.Thus (48) features the i, j-th chain of H . A chain (subchain) is closed if itselements add up to zero. As (1/

√N) · H is unitary, all its chains are closed.

It is not obvious, however, that any of these chains contain closed subchains.

Let us now construct a closed subchain pattern for H . For each chain ofH , let us split it, disjointly, into closed subchains:

(

Hi,k

(s)1

· Hj,k

(s)1

, Hi,k

(s)2

· Hj,k

(s)2

, . . . , Hi,k

(s)

p(s)

· Hj,k

(s)

p(s)

)

(49)where

s ∈ {1, 2, . . . , ri,j} designates subchains, ri,j being the number of closedsubchains the i, j-th chain is split into

⋃ri,j

s=1{k(s)1 , . . . , k

(s)

p(s)} = {1, 2, . . . , N}, p(s) being the length of the s-th sub-

chain

{k(s1)1 , . . . , k

(s1)

p(s1)} ∩ {k(s2)1 , . . . , k

(s2)

p(s2)} = ∅ if s1 6= s2

and splitting of {1, 2, . . . , N} into {k(s)1 , . . . , k

(s)

p(s)} is done independently for each

chain of H , that is the k-values above in fact depend also on i, j.A pattern according to which all the chains of H are split, in the above way,

into closed subchains, will be called a closed subchain pattern.

A closed subchain pattern may give rise to the affine Hadamard family,stemming from H , corresponding to this pattern. The space R, generating thisH(R) family (see Definition 3.4), is defined by the equations:

R1,j = 0 j ∈ {2, . . . , N} , (50)

Ri,1 = 0 i ∈ {1, . . . , N} , (51)

(as H(R) is made of dephased Hadamard matrices) and, for all 1 ≤ i < j ≤ Nand s ∈ {1, 2, . . . , ri,j} ,

(

Ri,k

(s)1

− Rj,k

(s)1

)

= . . . =

(

Ri,k

(s)

p(s)

− Rj,k

(s)

p(s)

)

, (52)

13

where the sets of indices {k(s)1 , . . . , k

(s)

p(s)} correspond to the considered pattern

splitting of the i, j-th chain into closed subchains (49). Recall that the sets ofk’s depend on i, j’s, ommited for simplicity of notation.

If the system (50,51,52) yields a nonzero space R, then H(R) is the affineHadamard family corresponding to the chosen closed subchain pattern. Actu-ally, any affine Hadamard family stemming from H is generated by some spaceR contained in a (probably larger) space R′, corresponding to some closed sub-chain pattern for H . The respective theorem and its proof will be published in[60].

It may happen that the system (50,51,52), for pattern P1(H), defines space

R, which is also obtained as the solution to (50,51,52) system shaped by

another pattern P2(H), imposing stronger conditions of type (52), as a result

of there being longer subchains in P2(H) composed of more than one subchain

of P1(H), for a given pair i, j. If this is not the case, we say that R is strictly

associated with pattern P1(H). Then the subchains of the i, j-th chain of

H(R) = H ◦ EXP(i · R), R ∈ R, distinguished according to pattern P1(H):

( Hi,k

(s)1

Hj,k

(s)1

exp(

i · ( Ri,k

(s)1

− Rj,k

(s)1

))

,

Hi,k

(s)2

Hj,k

(s)2

exp(

i · ( Ri,k

(s)2

− Rj,k

(s)2

))

,

. . . ,

Hi,k

(s)

p(s)

Hj,k

(s)

p(s)

exp

(

i · ( Ri,k

(s)

p(s)

− Rj,k

(s)

p(s)

)

)

) (53)

rotate independently as R runs along R satisfying (52).

Maximal affine Hadamard families stemming from H are generated by maxi-mal, in the sense of (42), solutions to (50,51,52) systems shaped by some specificclosed subchain patterns, which we also call maximal. We have been able tofind all of these for almost every complex Hadamard matrix considered in ourcatalogue. To our understanding, however, it becomes a serious combinatorialproblem already for N = 12. For example, for a real 12× 12 Hadamard matrix,each chain can be split in (6!)2 ways into two element closed subchains only.

Fortunately, as far as Fourier matrices are concerned, the allowed maximalsubchain patterns are especially regular. We are thus able to obtain all maximalaffine Hadamard families stemming from FN for an arbitrary N [60], as we havedone for N ≤ 16.

An alternative method of constructing affine Hadamard families, developedby Dita [22], is presented in section 4.5. We also refer the reader to the articleby Nicoara [20], in which conditions are given for the existence of one parame-ter families of commuting squares of finite dimensional von Neumann algebras.These conditions can be used to establish the existence of one parameter fam-ilies of complex Hadamard matrices, stemming from some H , which are notassumed to be affine.

14

4.3 Duplication of the matrix size

Certain ways of construction of real Hadamard matrices of an extended sizework also in the complex case. If A and B belong to HN then

H =

[

A BA −B

]

∈ H2N (54)

Furthermore, if A and B are taken to be in the dephased form, so is H .This method, originally due to Hadamard, can be generalized by realizingthat B can be multiplied at left by an arbitrary diagonal unitary matrix,E = diag(1, eid1, . . . , eidN−1). If A and B depend on a and b free parameters,respectively, then

H ′ =

[

A EBA −EB

]

(55)

represents an (a + b + N − 1)–parameter family of Hadamard matrices of size2N in the dephased form.

4.4 Quadruplication of the matrix size

In analogy to Eq. (54) one may quadruple the matrix size, in a constructionsimilar to that derived by Williamson [61] from quaternions. If A, B, C, D ∈ HN

are in the dephased form then

A B C DA −B C −DA B −C −DA −B −C D

∈ H4N (56)

and it has the dephased form. This form is preserved, if the blocks B, C andD are multiplied by diagonal unitary matrices E1, E2 and E3 respectively, eachcontaining unity and N − 1 free phases. Therefore (56) describes an [a + b +c + d + 3(N − 1)]–dimensional family of Hadamard matrices [22], where a, b, c, ddenote the number of free parameters contained in A, B, C, D, respectively.

4.5 Generalized method related to tensor product

It is not difficult to design a similar method which increases the size of aHadamard matrix by eight, but more generally, we can use tensor product6

to increase the size of the matrix K times. For any two Hadamard matrices,A ∈ HK and B ∈ HM , their tensor product A ⊗ B ∈ HKM . A more generalconstruction by Dita allows to use entire set of K (possibly different) Hadamardmatrices {B1, . . . , BK} of size M . Then the matrix

H =

A11B1 A12E2B2 . . A1KEKBK

. . . . .

. . . . .AK1B1 AK2E2B2 . . AKKEKBK

(57)

6Tensor products of Hadamard matrices of the Butson type were investigated in [39].

15

of size N = KM is Hadamard [22]. As before we introduce additional freephases by using K − 1 diagonal unitary matrices Ek. Each matrix depends onM − 1 phases, since the constraint [Ek]1,1 = 1 for k = 2, . . . , K is necessary topreserve the dephased form of H . Thus this orbit of (mostly) not equivalentHadamard matrices depends on

d = a +

K∑

j=1

bj + (K − 1)(M − 1) (58)

free parameters. Here a denotes the number of free parameters in A, while bj

denotes the number of free parameters in Bj . A similar construction givingat least (K − 1)(M − 1) free parameters was given by Haagerup [21]. In thesimplest case K ·M = 2 ·2 these methods give the standard 1–parameter N = 4family (65), while for K ·M = 2 ·3 one arrives with (2−1)(3−1) = 2 parameterfamily (69) of (mostly) inequivalent N = 6 Hadamard matrices.

The tensor product construction can work only for composite N , so it wasconjectured [29] that for a prime dimension N there exist only finitely manyinequivalent complex Hadamard matrices. However, this occurred to be falseafter a discovery by Petrescu [62], who found continuous families of complexHadamard matrices for certain prime dimensions. We are going to present hissolution for N = 7 and N = 13, while a similar construction [62] works alsofor N = 19, 31 and 79. For all primes N ≥ 7 there exist at least three isolatedcomplex Hadamard matrices, see [21], Thm. 3.10, p.320.

5 Catalogue of complex Hadamard matrices

In this section we list complex Hadamard matrices known to us. To save spacewe do not describe their construction but present a short characterization ofeach case.

Each entry H , provided in a dephased form, represents a continuous (2N −1)–dimensional family of matrices H obtained by multiplication by diagonalunitary matrices:

H(α1, ..., αN , β2, ..., βN ) = D1(α1, . . . , αN ) · H · D2(β2, . . . , βN ) (59)

where

D1(α1, . . . , αN ) = diag(eiα1 , . . . , eiαN )

D2(β2, . . . , βN) = diag(1, eiβ2 , . . . , eiβN )

Furthermore, each H in the list represents matrices obtained by discretepermutations of its rows and columns, and equivalent in the sense of (5). Fora given size N we enumerate families by capital letters associated with a givenconstruction. The superscript in brackets denotes the dimension of an orbit. For

instance, F(2)6 represents the two–parameter family of N = 6 complex Hadamard

matrices stemming from the Fourier matrix F6. Displaying continuous familiesof Hadamard matrices we shall use the symbol • to denote zeros in phase vari-ation matrices. For completeness we shall start with a trivial case.

16

5.1 N = 1

F(0)1 = F1 = [1].

5.2 N = 2

All complex 2 × 2 Hadamard matrices are equivalent to the Fourier matrix F2

([21], Prop.2.1, p.298):

F2 = F(0)2 =

[

1 11 −1

]

. (60)

The complex Hadamard matrix F2 is isolated.

The set of inequivalent Hadamard matrices of size 2 contains one elementonly, G2 = {F2}.

5.3 N = 3

All complex 3 × 3 Hadamard matrices are equivalent to the Fourier matrix F3

([21], Prop.2.1, p.298):

F3 = F(0)3 =

1 1 11 w w2

1 w2 w

, (61)

where w = exp(i · 2π/3), so w3 = 1.F3 is an isolated complex Hadamard matrix.

The set of dephased representatives can be taken as G3 = {F3}.

5.4 N = 4

Every 4 × 4 complex Hadamard matrix is equivalent to a matrix belonging to

the only maximal affine Hadamard family F(1)4 (a) stemming from F4 [21]. The

F(1)4 (a) family is given by the formula:

F(1)4 (a) = F4 ◦ EXP(i · R

F(1)4

(a)) (62)

where

F4 =

1 1 1 11 w w2 w3

1 w2 1 w2

1 w3 w2 w

=

1 1 1 11 i −1 −i1 −1 1 −11 −i −1 i

(63)

17

where w = exp(i · 2π/4) = i, so w4 = 1, w2 = −1, and

RF

(1)4

(a) =

• • • •• a • a• • • •• a • a

. (64)

Thus the orbit stemming from F4 reads:

F(1)4 (a) =

1 1 1 11 i · eia −1 −i · eia

1 −1 1 −11 −i · eia −1 i · eia

. (65)

The above orbit is permutation equivalent to:

F(1)4 (α) =

[

[F2]1,1 · F2 [F2]1,2 · D(α) · F2

[F2]2,1 · F2 [F2]2,2 · D(α) · F2

]

, (66)

where D(α) is the 2 × 2 diagonal matrix diag(1, eiα).This orbit is constructed with the Dita’s method [22], by setting K = M =

2, A = B1 = B2 = F2, E2 = D(α) in (57).

It passes through a permuted F4:

F4 = F(1)4 (π/2) · [e1, e3, e2, e4]T (67)

where ei is the i-th standard basis column vector, and through F2⊗F2 = F(1)4 (0).

Note that F4 and F2 ⊗ F2 are, according to [59], inequivalent.

The F(1)4 orbit of (62) is symmetric, so it is self–cognate. Replacing a by

a + π yields F(1)4 (a) with the 2-nd and 4-th column exchanged, so F

(1)4 (a) ≃

F(1)4 (a + π).

The set of dephased representatives can be taken as G4 = {F (1)4 (a) : a ∈

[0, π)}

5.5 N = 5

As shown by Haagerup in [21] (Th. 2.2, p. 298) for N = 5 all complex Hadamardmatrices are equivalent to the Fourier matrix F5:

F5 = F(0)5 =

1 1 1 1 11 w w2 w3 w4

1 w2 w4 w w3

1 w3 w w4 w2

1 w4 w3 w2 w

, (68)

18

where w = exp(i · 2π/5) so w5 = 1.The above matrix F5 is isolated.

The set of dephased representatives can be taken as G5 = {F5}.

5.6 N = 6

5.6.1 Orbits stemming from F6

The only maximal affine Hadamard families stemming from F6 are:

F(2)6 (a, b) = F6 ◦ EXP

(

i · RF

(2)6

(a, b))

(69)

(

F(2)6 (a, b)

)T

= F6 ◦ EXP

(

i ·(

RF

(2)6

(a, b))T)

(70)

where

F6 =

1 1 1 1 1 11 w w2 w3 w4 w5

1 w2 w4 1 w2 w4

1 w3 1 w3 1 w3

1 w4 w2 1 w4 w2

1 w5 w4 w3 w2 w

(71)

where w = exp(i · 2π/6) so w6 = 1, w3 = −1, and

RF

(2)6

(a, b) =

• • • • • •• a b • a b• • • • • •• a b • a b• • • • • •• a b • a b

(72)

The above families (69) and (70) are cognate. The defect (see Def. ) reads

d(F6) = 4.At least one of the above orbits is permutation equivalent to one of the orbits

obtained using the Dita’s method, either

F(2)6A (α1, α2) =

[F3]1,1 · F2 [F3]1,2 · D(α1) · F2 [F3]1,3 · D(α2) · F2

[F3]2,1 · F2 [F3]2,2 · D(α1) · F2 [F3]2,3 · D(α2) · F2

[F3]3,1 · F2 [F3]3,2 · D(α1) · F2 [F3]3,3 · D(α2) · F2

(73)

where D(α) is the 2 × 2 diagonal matrix diag(1, eiα), by setting K = 3 , M =2, A = F3, B1 = B2 = B3 = F2, E2 = D(α1), E3 = D(α2) in (57),

19

or

F(2)6B (β1, β2) =

[

[F2]1,1 · F3 [F2]1,2 · D(β1, β2) · F3

[F2]2,1 · F3 [F2]2,2 · D(β1, β2) · F3

]

(74)

where D(β1, β2) is the 3 × 3 diagonal matrix diag(1, eiβ1, eiβ2), by setting K =2, M = 3, A = F2, B1 = B2 = F3, E2 = D(β1, β2) in (57).

The above statement is true because the orbits (73) and (74) pass through

F(2)6A (0, 0) = F3 ⊗ F2 and F

(2)6B (0, 0) = F2 ⊗ F3, both of which are, accord-

ing to [59], permutation equivalent to F6. Thus (73) and (74) are maximalaffine Hadamard families stemming from permuted F6’s. These orbits wereconstructed in the work of Haagerup [21].

A change of the phase a by π in the family F(2)6 corresponds to the exchange

of the 2-nd and 5-th column of F(2)6 (a, b), while a change of b by π is equiv-

alent to the exchange of the 3-rd and 6-th column of F(2)6 (a, b). This implies

(permutation) equivalence relation:

F(2)6 (a, b) ≃ F

(2)6 (a + π, b) ≃ F

(2)6 (a, b + π) ≃ F

(2)6 (a + π, b + π) . (75)

5.6.2 1-parameter orbits

There are precisely five permutation equivalent 1-parameter maximal affineHadamard families stemming from the symmetric matrix D6:

D6 =

1 1 1 1 1 11 −1 i −i −i i1 i −1 i −i −i1 −i i −1 i −i1 −i −i i −1 i1 i −i −i i −1

(76)

namely

D(1)6 (c) = D6 ◦ EXP

(

i · RD

(1)6

(c))

(77)

where

RD

(1)6

(c) =

• • • • • •• • • • • •• • • c c •• • −c • • −c• • −c • • −c• • • c c •

(78)

20

and

P1 · D(1)6 (c) · PT

1 = D6 ◦ EXP(

i · P1RD(1)6

(c)PT1

)

(79)

P2 · D(1)6 (c) · PT

2 = D6 ◦ EXP(

i · P2RD(1)6

(c)PT2

)

(80)

P3 · D(1)6 (c) · PT

3 = D6 ◦ EXP(

i · P3RD(1)6

(c)PT3

)

(81)

P4 · D(1)6 (c) · PT

4 = D6 ◦ EXP(

i · P4RD(1)6

(c)PT4

)

(82)

where

P1 = [e1, e3, e2, e6, e5, e4] (83)

P2 = [e1, e4, e3, e2, e6, e5] (84)

P3 = [e1, e6, e2, e3, e4, e5] (85)

P4 = [e1, e5, e4, e3, e2, e6] (86)

and ei denotes the i-th standard basis column vector.

We have the permutation equivalence

D(1)6 (c + π) = PT

− · D(1)6 (c) · P− for P− = [e1, e2, e3, e5, e4, e6] (87)

Also

D(1)6 (−c) =

(

D(1)6 (c)

)T

(88)

thus {D(1)6 (c) : c ∈ [0, 2π)} and {

(

D(1)6 (c)

)T

: c ∈ [0, 2π)} are equal sets, that

is D(1)6 is a self–cognate family

None of the matrices of F(2)6 and

(

F(2)6

)T

of (69) and (70) are equivalent to

any of the matrices of D(1)6 .

Obviously, the above remarks apply to the remaining orbits stemming fromD6.

The D(1)6 orbit was presented in [22] in the Introduction, and its ’starting

point’ matrix D6 of (76) even in the earlier work [21] p.307 (not dephased).

5.6.3 The ’cyclic 6–roots’ matrix

There exists another, inequivalent to any of the above 6 × 6 matrices, complexHadamard matrix derived in [21] from the results of [32] on so called cyclic6–roots.

The matrix C(0)6 below is circulant, i.e. it has the structure

[

C(0)6

]

i,j=

x(i−j mod 6)+1, where

x = [1, i/d, −1/d, −i, −d, id]T (89)

21

and

d =1 −

√3

2+ i ·

(√3

2

)12

(90)

is a root of the equation d2 − (1 −√

3)d + 1 = 0.

The matrix C(0)6 and its dephased form C

(0)6 read:

C(0)6 =

1 i d −d −i −d−1 id−1

id−1 1 i d −d −i −d−1

−d−1 id−1 1 i d −d −i−i −d−1 id−1 1 i d −d−d −i −d−1 id−1 1 i di d −d −i −d−1 id−1 1

(91)

C(0)6 =

1 1 1 1 1 11 −1 −d −d2 d2 d1 −d−1 1 d2 −d3 d2

1 −d−2 d−2 −1 d2 −d2

1 d−2 −d−3 d−2 1 −d1 d−1 d−2 −d−2 −d−1 −1

(92)

The circulant structure of C(0)6 implies that it is equivalent to

(

C(0)6

)T

, so

C(0)6 ≃

(

C(0)6

)T

(see (47)). Thus C(0)6 ≃ C

(0)6 ( ⇐⇒ C

(0)6 ≃ C

(0)6 ).

No affine Hadamard family stems from C(0)6 . However, we do not know if

the cyclic 6–roots matrix is isolated since the defect d(C(0)6 ) = 4 and we cannot

exclude existence of some other orbit.

5.6.4 The ’spectral set’ 6 × 6 matrix

Another complex Hadamard matrix found by Tao [35] plays an important rolein investigation of spectral sets and disproving the Fuglede’s conjecture [36, 38].

It is a symmetric matrix S(0)6 , which belongs to the Butson class H(3, 6), so its

entries depend on the third root of unity ω = exp(i · 2π/3),

S(0)6 =

1 1 1 1 1 11 1 ω ω ω2 ω2

1 ω 1 ω2 ω2 ω1 ω ω2 1 ω ω2

1 ω2 ω2 ω 1 ω1 ω2 ω ω2 ω 1

. (93)

22

Thus the corresponding log-Hadamard matrix reads

ΦS6 =2π

3

0 0 0 0 0 00 0 1 1 2 20 1 0 2 2 10 1 2 0 1 20 2 2 1 0 10 2 1 2 1 0

. (94)

Its defect is equal to zero, d(S6) = 0, hence the matrix (93) is isolated.Spectral sets allow to construct certain Hadamard matrices for other compositedimensions (see prop. 2.2. in [37]), but it is not yet established, in which casesthis method yields new solutions.

5.7 N = 7

5.7.1 Orbits stemming from F7

F7 is an isolated 7 × 7 complex Hadamard matrix:

F7 = F(0)7 =

1 1 1 1 1 1 11 w w2 w3 w4 w5 w6

1 w2 w4 w6 w w3 w5

1 w3 w6 w2 w5 w w4

1 w4 w w5 w2 w6 w3

1 w5 w3 w w6 w4 w2

1 w6 w5 w4 w3 w2 w

, (95)

where w = exp(i · 2π/7), so w7 = 1.

5.7.2 1-parameter orbits

There are precisely three permutation equivalent 1-parameter maximal affineHadamard families stemming from the symmetric matrix being a permuted’starting point’ for the 1-parameter orbit found by Petrescu [62]:

P7 =

1 1 1 1 1 1 11 w w4 w5 w3 w3 w1 w4 w w3 w5 w3 w1 w5 w3 w w4 w w3

1 w3 w5 w4 w w w3

1 w3 w3 w w w4 w5

1 w w w3 w3 w5 w4

, (96)

where w = exp(i · 2π/6), so w6 = 1, w3 = −1.

23

They are:

P(1)7 (a) = P7 ◦ EXP

(

i · RP

(1)7

(a))

(97)

where

RP

(1)7

(a) =

• • • • • • •• a a • • • •• a a • • • •• • • −a −a • •• • • −a −a • •• • • • • • •• • • • • • •

(98)

and

P1 · P (1)7 (a) · PT

2 = P7 ◦ EXP(

i · P1RP(1)7

(a)PT2

)

(99)

P2 · P (1)7 (a) · PT

1 = P7 ◦ EXP(

i · P2RP(1)7

(a)PT1

)

(100)

where

P1 = [e1, e7, e3, e4, e6, e5, e2] P2 = [e1, e2, e7, e6, e5, e4, e3] (101)

and ei denotes the i-th standard basis column vector.

The above orbits are permutation equivalent to the 7 × 7 family found by

Petrescu [62]. They all are cognate, and the P(1)7 orbit is even self–cognate since

P(1)7 (a) =

(

P(1)7 (a)

)T

. (102)

Also

P(1)7 (−a) = PT · P (1)

7 (a) · P for P = [e1, e4, e5, e2, e3, e7, e6] (103)

so P(1)7 (−a) ≃ P

(1)7 (a)

Due to some freedom in the construction of family components the methodof Petrescu allows one to build other families of Hadamard matrices similar toP

(1)7 . Not knowing if they are inequivalent we are not going to consider them

here.

5.7.3 The ’cyclic 7–roots’ matrices

There exist only four inequivalent 7× 7 complex Hadamard matrices, inequiva-lent to F7 and the 1-parameter family found by Petrescu (see previous subsec-tion), associated with nonclassical cyclic 7–roots. This result was obtained in[21] and is based on the catalogue of all cyclic 7–roots presented in [32].

24

The two matrices C(0)7A , C

(0)7B correspond to the so–called ’index 2’ solu-

tions to the cyclic 7–roots problem. They have the circulant structure [U ]i,j =x(i−j mod 7)+1, where

x = [1, 1, 1, d, 1, d, d] for C(0)7A (104)

x = [1, 1, 1, d, 1, d, d] for C(0)7B (105)

and

d =−3 + i

√7

4such that d2 +

3

2d + 1 = 0 =⇒ d · d = 1 (106)

The corresponding dephased matrices are denoted as C(0)7A and C

(0)7B :

C(0)7A =

1 d d 1 d 1 11 1 d d 1 d 11 1 1 d d 1 dd 1 1 1 d d 11 d 1 1 1 d dd 1 d 1 1 1 dd d 1 d 1 1 1

C(0)7A =

1 1 1 1 1 1 11 d−1 1 d d−1 d 11 d−1 d−1 d 1 1 d1 d−2 d−2 d−1 d−1 1 d−1

1 1 d−1 1 d−1 d d1 d−2 d−1 d−1 d−2 d−1 11 d−1 d−2 1 d−2 d−1 d−1

(107)

C(0)7B =

1 d−1 d−1 1 d−1 1 11 1 d−1 d−1 1 d−1 11 1 1 d−1 d−1 1 d−1

d−1 1 1 1 d−1 d−1 11 d−1 1 1 1 d−1 d−1

d−1 1 d−1 1 1 1 d−1

d−1 d−1 1 d−1 1 1 1

C(0)7B =

1 1 1 1 1 1 11 d 1 d−1 d d−1 11 d d d−1 1 1 d−1

1 d2 d2 d d 1 d1 1 d 1 d d−1 d−1

1 d2 d d d2 d 11 d d2 1 d2 d d

(108)

There holds C(0)7B = C

(0)7A ( ⇐⇒ C

(0)7B = C

(0)7A ) and the matrices C

(0)7A ,

C(0)7A and C

(0)7B , C

(0)7B are equivalent to

(

C(0)7A

)T

,(

C(0)7A

)T

and(

C(0)7B

)T

,(

C(0)7B

)T

respectively (see (47)).

The structure of C(0)7C , C

(0)7D related to ’index 3’ solutions to the cyclic 7–roots

problem is again [U ]i,j = x(i−j mod 7)+1, where x is a bit more complicated.We put:

x = [1, A, B, C, C, B, A]T for C(0)7C (109)

x = [1, A, B, C, C, B, A]T for C(0)7D (110)

25

where A = a, B = ab, C = abc are products of algebraic numbers a, b, cof modulus equal to 1 (see the final remark 3.11 of [21]). Their numericalapproximations are given by

a ≈ exp(i · 4.312839) (111)

b ≈ exp(i · 1.356228) (112)

c ≈ exp(i · 1.900668) (113)

(114)

and then

A = a ≈ (−0.389004) + i · (−0.921236) (115)

B = ab ≈ (0.817282) + i · (−0.576238) (116)

C = abc ≈ (0.280434) + i · (0.959873) (117)

Again, C(0)7C , C

(0)7D denote the dephased versions of C

(0)7C , C

(0)7D , which read:

C(0)7C =

1 A B C C B AA 1 A B C C BB A 1 A B C CC B A 1 A B CC C B A 1 A BB C C B A 1 AA B C C B A 1

C(0)7D =

1 A−1 B−1 C−1 C−1 B−1 A−1

A−1 1 A−1 B−1 C−1 C−1 B−1

B−1 A−1 1 A−1 B−1 C−1 C−1

C−1 B−1 A−1 1 A−1 B−1 C−1

C−1 C−1 B−1 A−1 1 A−1 B−1

B−1 C−1 C−1 B−1 A−1 1 A−1

A−1 B−1 C−1 C−1 B−1 A−1 1

(118)

Then

C(0)7C =

1 1 1 1 1 1 11 A−2 B−1 A−1BC−1 A−1 A−1B−1C A−2B1 B−1 B−2 AB−1C−1 C−1 B−2C A−1B−1C1 A−1BC−1 AB−1C−1 C−2 AC−2 C−1 A−1

1 A−1 C−1 AC−2 C−2 AB−1C−1 A−1BC−1

1 A−1B−1C B−2C C−1 AB−1C−1 B−2 B−1

1 A−2B A−1B−1C A−1 A−1BC−1 B−1 A−2

=

1 1 1 1 1 1 11 a−2 a−1b−1 a−1c−1 a−1 a−1c a−1b1 a−1b−1 a−2b−2 a−1b−2c−1 a−1b−1c−1 a−1b−1c a−1c1 a−1c−1 a−1b−2c−1 a−2b−2c−2 a−1b−2c−2 a−1b−1c−1 a−1

1 a−1 a−1b−1c−1 a−1b−2c−2 a−2b−2c−2 a−1b−2c−1 a−1c−1

1 a−1c a−1b−1c a−1b−1c−1 a−1b−2c−1 a−2b−2 a−1b−1

1 a−1b a−1c a−1 a−1c−1 a−1b−1 a−2

(119)

26

and

C(0)7D =

1 1 1 1 1 1 11 A2 B AB−1C A ABC−1 A2B−1

1 B B2 A−1BC C B2C−1 ABC−1

1 AB−1C A−1BC C2 A−1C2 C A1 A C A−1C2 C2 A−1BC AB−1C1 ABC−1 B2C−1 C A−1BC B2 B1 A2B−1 ABC−1 A AB−1C B A2

=

1 1 1 1 1 1 11 a2 ab ac a ac−1 ab−1

1 ab a2b2 ab2c abc abc−1 ac−1

1 ac ab2c a2b2c2 ab2c2 abc a1 a abc ab2c2 a2b2c2 ab2c ac1 ac−1 abc−1 abc ab2c a2b2 ab1 ab−1 ac−1 a ac ab a2

(120)

The matrices C(0)7C and C

(0)7D are symmetric and are related by complex con-

jugation, C(0)7D = C

(0)7C . All four cyclic 7–roots Hadamard matrices are isolated

[20].

5.8 N = 8

5.8.1 Orbits stemming from F8

The only maximal affine Hadamard family stemming from F8 is the 5-parameterorbit:

F(5)8 (a, b, c, d, e) = F8 ◦ EXP(i · R

F(5)8

(a, b, c, d, e)) (121)

where

F8 =

1 1 1 1 1 1 1 11 w w2 w3 w4 w5 w6 w7

1 w2 w4 w6 1 w2 w4 w6

1 w3 w6 w w4 w7 w2 w5

1 w4 1 w4 1 w4 1 w4

1 w5 w2 w7 w4 w w6 w3

1 w6 w4 w2 1 w6 w4 w2

1 w7 w6 w5 w4 w3 w2 w

, (122)

27

where w = exp(i · 2π/8), so w8 = 1, w4 = −1, w2 = i, and

RF

(5)8

(a, b, c, d, e) =

• • • • • • • •• a b c • a b c• d • d • d • d• e b c − a + e • e b c − a + e• • • • • • • •• a b c • a b c• d • d • d • d• e b c − a + e • e b c − a + e

. (123)

The family F(5)8 is self–cognate.

The above orbit is permutation equivalent to the orbit constructed with theDita’s method [22]:

F(5)8 (α1, . . . , α5) =

[

[F2]1,1 · F(1)4 (α1) [F2]1,2 · D(α3, α4, α5) · F (1)

4 (α2)

[F2]2,1 · F(1)4 (α1) [F2]2,2 · D(α3, α4, α5) · F (1)

4 (α2)

]

(124)where

F(1)4 (α) =

1 1 1 11 ieiα −1 −ieiα

1 −1 1 −11 −ieiα −1 ieiα

(125)

is the only maximal affine Hadamard family stemming from F4 and D(α3, α4, α5)is the 4 × 4 diagonal matrix diag(1, eiα3 , eiα4 , eiα5).

Eq. (124) leads to F2 ⊗ F4 for α1 = . . . = α5 = 0, which is not equivalent

to F8, see [59]. However, F(5)8 (0, 0, (1/8)2π, (2/8)2π, (3/8)2π) is permutation

equivalent to F8, since

F8 = F(5)8 (0, 0, (1/8)2π, (2/8)2π, (3/8)2π) · [e1, e3, e5, e7, e2, e4, e6, e8]T , (126)

where ei denotes the i-th column vector of the standard basis of C8. ThusEq. (124) generates the only maximal affine Hadamard family stemming frompermuted F8.

The matrix F(5)8 (π/2, π/2, 0, 0, 0) yields the only real 8×8 Hadamard matrix,

up to permutations and multiplying rows and columns by −1. It is dephased,so it is permutation equivalent to F2 ⊗ F2 ⊗ F2.

Therefore all appropriately permuted tensor products of Fourier matrices,F2 ⊗ F2 ⊗ F2, F2 ⊗ F4 and F8, although inequivalent [59], are connected by theorbit (124).

28

5.9 N = 9

5.9.1 Orbits stemming from F9

The only maximal affine Hadamard family stemming from F9 is the 4-parameterorbit:

F(4)9 (a, b, c, d) = F9 ◦ EXP(i · R

F(4)9

(a, b, c, d)) , (127)

where

F9 =

1 1 1 1 1 1 1 1 11 w w2 w3 w4 w5 w6 w7 w8

1 w2 w4 w6 w8 w w3 w5 w7

1 w3 w6 1 w3 w6 1 w3 w6

1 w4 w8 w3 w7 w2 w6 w w5

1 w5 w w6 w2 w7 w3 w8 w4

1 w6 w3 1 w6 w3 1 w6 w3

1 w7 w5 w3 w w8 w6 w4 w2

1 w8 w7 w6 w5 w4 w3 w2 w

, (128)

with w = exp(i · 2π/9), so w9 = 1, and

RF

(4)9

(a, b, c, d) =

• • • • • • • • •• a b • a b • a b• c d • c d • c d• • • • • • • • •• a b • a b • a b• c d • c d • c d• • • • • • • • •• a b • a b • a b• c d • c d • c d

. (129)

The orbit F(4)9 is self–cognate. Observe that its dimension is equal to the defect,

d(F9) = 4, which follows from Eq. (19).

It is permutation equivalent to the 4-dimensional orbit passing through apermuted F9, constructed using the Dita’s method:

F(4)9 (α1, . . . , α4) =

[F3]1,1 · F3 [F3]1,2 · D(α1, α2) · F3 [F3]1,3 · D(α3, α4) · F3

[F3]2,1 · F3 [F3]2,2 · D(α1, α2) · F3 [F3]2,3 · D(α3, α4) · F3

[F3]3,1 · F3 [F3]3,2 · D(α1, α2) · F3 [F3]3,3 · D(α3, α4) · F3

(130)where D(α, β) is the 3 × 3 diagonal matrix diag(1, eiα, eiβ).

The matrix F(4)9 (0, 0, 0, 0) = F3 ⊗ F3 is not equivalent to F9 [59], but

F9 = F(4)9 ((1/9)2π, (2/9)2π, (2/9)2π, (4/9)2π) · [e1, e4, e7, e2, e5, e8, e3, e6, e9]T ,

(131)

29

where ei are the standard basis column vectors.Thus both inequivalent permuted matrices, F3 ⊗ F3 and F9, are connected

by the orbit (130).

5.10 N = 10

5.10.1 Orbits stemming from F10

The only maximal affine Hadamard families stemming from F10 are:

F(4)10 (a, b, c, d) = F10 ◦ EXP

(

i · RF

(4)10

(a, b, c, d))

(132)

(

F(4)10 (a, b, c, d)

)T

= F10 ◦ EXP

(

i ·(

RF

(4)10

(a, b, c, d))T)

, (133)

where

F10 =

1 1 1 1 1 1 1 1 1 11 w w2 w3 w4 w5 w6 w7 w8 w9

1 w2 w4 w6 w8 1 w2 w4 w6 w8

1 w3 w6 w9 w2 w5 w8 w w4 w7

1 w4 w8 w2 w6 1 w4 w8 w2 w6

1 w5 1 w5 1 w5 1 w5 1 w5

1 w6 w2 w8 w4 1 w6 w2 w8 w4

1 w7 w4 w w8 w5 w2 w9 w6 w3

1 w8 w6 w4 w2 1 w8 w6 w4 w2

1 w9 w8 w7 w6 w5 w4 w3 w2 w

(134)

with w = exp(i · 2π/10), so w10 = 1, w5 = −1, and

RF

(4)10

(a, b, c, d) =

• • • • • • • • • •• a b c d • a b c d• • • • • • • • • •• a b c d • a b c d• • • • • • • • • •• a b c d • a b c d• • • • • • • • • •• a b c d • a b c d• • • • • • • • • •• a b c d • a b c d

(135)

The affine Hadamard families F(4)10 and

(

F(4)10

)T

are cognate.

At least one of them must be permutation equivalent to an orbit constructedusing the Dita’s method, either:

F(4)10A(α1, . . . , α4) =

[

[F2]1,1 · F5 [F2]1,2 · D(α1, . . . , α4) · F5

[F2]2,1 · F5 [F2]2,2 · D(α1, . . . , α4) · F5

]

(136)

30

where D(α1, . . . , α4) is the 5 × 5 diagonal matrix diag(1, eiα1 , . . . , eiα4), or

F(4)10B(β1, . . . , β4) (137)

such that its i, j-th 2×2 block is equal to [F5]i,j ·D(α)·F2 , where i, j ∈ {1 . . . 5},

D(α) is the diagonal matrix diag(1, eiα) and α = 0, β1, . . . , β4 for j = 1, 2, . . . , 5respectively.

This is because F(4)10A(0) = F2 ⊗ F5 and F

(4)10B(0) = F5 ⊗ F2 are, according

to [59], permutation equivalent to F10, so both Dita’s orbits are maximal affineHadamard families stemming from permuted F10’s.

5.11 N = 11

5.11.1 Orbits stemming from F11

The Fourier matrix F11 is an isolated 11 × 11 complex Hadamard matrix:

F11 = F(0)11 =

1 1 1 1 1 1 1 1 1 1 11 w w2 w3 w4 w5 w6 w7 w8 w9 w10

1 w2 w4 w6 w8 w10 w w3 w5 w7 w9

1 w3 w6 w9 w w4 w7 w10 w2 w5 w8

1 w4 w8 w w5 w9 w2 w6 w10 w3 w7

1 w5 w10 w4 w9 w3 w8 w2 w7 w w6

1 w6 w w7 w2 w8 w3 w9 w4 w10 w5

1 w7 w3 w10 w6 w2 w9 w5 w w8 w4

1 w8 w5 w2 w10 w7 w4 w w9 w6 w3

1 w9 w7 w5 w3 w w10 w8 w6 w4 w2

1 w10 w9 w8 w7 w6 w5 w4 w3 w2 w

(138)where w = exp(i · 2π/11), so w11 = 1.

5.11.2 ’Cyclic 11–roots’ matrices

There are precisely two ≃ equivalence classes of complex Hadamard matrices,inequivalent to F11, associated with the so–called nonclassical ’index 2’ cyclic11–roots. This result is drawn in [21] p.319.

The classes are represented by the matrices C(0)11A, C

(0)11B below, their re-

spective dephased forms are denoted by C(0)11A, C

(0)11B. Both matrices have the

circulant structure [U ]i,j = x(i−j mod 11)+1, where

x = [1, 1, e, 1, 1, 1, e, e, e, 1, e] for C(0)11A (139)

x = [1, 1, e, 1, 1, 1, e, e, e, 1, e] for C(0)11B (140)

31

and

e = −5

6+ i ·

√11

6(141)

.There holds

C(0)11B = C

(0)11A so C

(0)11B = C

(0)11A (142)

Applying transposition to C(0)11A, C

(0)11B – equivalently to C

(0)11A, C

(0)11B – yields

a matrix equivalent to the original one (see (47)).The matrices are given by:

C(0)11A =

1 e 1 e e e 1 1 1 e 11 1 e 1 e e e 1 1 1 ee 1 1 e 1 e e e 1 1 11 e 1 1 e 1 e e e 1 11 1 e 1 1 e 1 e e e 11 1 1 e 1 1 e 1 e e ee 1 1 1 e 1 1 e 1 e ee e 1 1 1 e 1 1 e 1 ee e e 1 1 1 e 1 1 e 11 e e e 1 1 1 e 1 1 ee 1 e e e 1 1 1 e 1 1

, (143)

C(0)11B =

1 e−1 1 e−1 e−1 e−1 1 1 1 e−1 11 1 e−1 1 e−1 e−1 e−1 1 1 1 e−1

e−1 1 1 e−1 1 e−1 e−1 e−1 1 1 11 e−1 1 1 e−1 1 e−1 e−1 e−1 1 11 1 e−1 1 1 e−1 1 e−1 e−1 e−1 11 1 1 e−1 1 1 e−1 1 e−1 e−1 e−1

e−1 1 1 1 e−1 1 1 e−1 1 e−1 e−1

e−1 e−1 1 1 1 e−1 1 1 e−1 1 e−1

e−1 e−1 e−1 1 1 1 e−1 1 1 e−1 11 e−1 e−1 e−1 1 1 1 e−1 1 1 e−1

e−1 1 e−1 e−1 e−1 1 1 1 e−1 1 1

(144)

C(0)11A =

1 1 1 1 1 1 1 1 1 1 11 e−1 e e−1 1 1 e 1 1 e−1 e1 e−2 e−1 e−1 e−2 e−1 1 1 e−1 e−2 e−1

1 1 1 e−1 1 e−1 e e e e−1 11 e−1 e e−1 e−1 1 1 e e 1 11 e−1 1 1 e−1 e−1 e 1 e 1 e1 e−2 e−1 e−2 e−1 e−2 e−1 1 e−1 e−1 11 e−1 e−1 e−2 e−2 e−1 e−1 e−1 1 e−2 11 e−1 1 e−2 e−2 e−2 1 e−1 e−1 e−1 e−1

1 1 e 1 e−1 e−1 1 e 1 e−1 e1 e−2 1 e−1 e−1 e−2 e−1 e−1 1 e−2 e−1

(145)

32

C(0)11B =

1 1 1 1 1 1 1 1 1 1 11 e e−1 e 1 1 e−1 1 1 e e−1

1 e2 e e e2 e 1 1 e e2 e1 1 1 e 1 e e−1 e−1 e−1 e 11 e e−1 e e 1 1 e−1 e−1 1 11 e 1 1 e e e−1 1 e−1 1 e−1

1 e2 e e2 e e2 e 1 e e 11 e e e2 e2 e e e 1 e2 11 e 1 e2 e2 e2 1 e e e e1 1 e−1 1 e e 1 e−1 1 e e−1

1 e2 1 e e e2 e e 1 e2 e

(146)

5.11.3 Nicoara’s 11 × 11 complex Hadamard matrix

Another equivalence class of 11 × 11 Hadamard matrices is represented by thematrix communicated to the authors by Nicoara:

N(0)11 =

1 1 1 1 1 1 1 1 1 1 11 a −a −a −a −1 −1 −a−1 −a−1 −1 −11 −a a −a −a −1 −1 −1 −1 −a−1 −a−1

1 −a −a −1 a −1 −a −a−1 −1 −a−1 −11 −a −a a −1 −a −1 −1 −a−1 −1 −a−1

1 −1 −1 −1 −a −a 1 −a−1 −1 −1 −a−1

1 −1 −1 −a −1 1 −a −1 −a−1 −a−1 −11 −a−1 −1 −a−1 −1 −a−1 −1 −a−1 a−1 −a−2 −a−2

1 −a−1 −1 −1 −a−1 −1 −a−1 a−1 −a−1 −a−2 −a−2

1 −1 −a−1 −a−1 −1 −1 −a−1 −a−2 −a−2 −a−2 a−2

1 −1 −a−1 −1 −a−1 −a−1 −1 −a−2 −a−2 a−2 −a−2

(147)where

a = −3

4− i ·

√7

4(148)

The above matrix is isolated7 since its defect d(N(0)11 ) = 0.

7R.Nicoara, private communication

33

5.12 N = 12

5.12.1 Orbits stemming from F12

The only maximal affine Hadamard families stemming from F12 are:

F(9)12A(a, b, c, d, e, f, g, h, i) = F12 ◦ EXP

(

i · RF

(9)12A

(a, . . . , i))

(149)

F(9)12B(a, b, c, d, e, f, g, h, i) = F12 ◦ EXP

(

i · RF

(9)12B

(a, . . . , i))

(150)

F(9)12C(a, b, c, d, e, f, g, h, i) = F12 ◦ EXP

(

i · RF

(9)12C

(a, . . . , i))

(151)

F(9)12D(a, b, c, d, e, f, g, h, i) = F12 ◦ EXP

(

i · RF

(9)12D

(a, . . . , i))

(152)

(

F(9)12B(a, b, c, d, e, f, g, h, i)

)T

= F12 ◦ EXP

(

i ·(

RF

(9)12B

(a, . . . , i))T)

(

F(9)12C(a, b, c, d, e, f, g, h, i)

)T

= F12 ◦ EXP

(

i ·(

RF

(9)12C

(a, . . . , i))T)

(

F(9)12D(a, b, c, d, e, f, g, h, i)

)T

= F12 ◦ EXP

(

i ·(

RF

(9)12D

(a, . . . , i))T)

where

F12 =

1 1 1 1 1 1 1 1 1 1 1 11 w w2 w3 w4 w5 w6 w7 w8 w9 w10 w11

1 w2 w4 w6 w8 w10 1 w2 w4 w6 w8 w10

1 w3 w6 w9 1 w3 w6 w9 1 w3 w6 w9

1 w4 w8 1 w4 w8 1 w4 w8 1 w4 w8

1 w5 w10 w3 w8 w w6 w11 w4 w9 w2 w7

1 w6 1 w6 1 w6 1 w6 1 w6 1 w6

1 w7 w2 w9 w4 w11 w6 w w8 w3 w10 w5

1 w8 w4 1 w8 w4 1 w8 w4 1 w8 w4

1 w9 w6 w3 1 w9 w6 w3 1 w9 w6 w3

1 w10 w8 w6 w4 w2 1 w10 w8 w6 w4 w2

1 w11 w10 w9 w8 w7 w6 w5 w4 w3 w2 w

(153)

34

where w = exp(i · 2π/12), so w12 = 1, w6 = −1, w3 = i, and

RF

(9)12A

(a, . . . , i) =

• • • • • • • • • • • •• a b c d e • a b c d e• f • f • f • f • f • f• g b c − a + g d e − a + g • g b c − a + g d e − a + g• h • h • h • h • h • h• i b c − a + i d e − a + i • i b c − a + i d e − a + i• • • • • • • • • • • •• a b c d e • a b c d e• f • f • f • f • f • f• g b c − a + g d e − a + g • g b c − a + g d e − a + g• h • h • h • h • h • h• i b c − a + i d e − a + i • i b c − a + i d e − a + i

(154)

RF

(9)12B

(a, . . . , i) =

• • • • • • • • • • • •• a b c d e • a b c d e• f g • f g • f g • f g• h i c d − a + h e − b + i • h i c d − a + h e − b + i• • • • • • • • • • • •• a b c d e • a b c d e• f g • f g • f g • f g• h i c d − a + h e − b + i • h i c d − a + h e − b + i• • • • • • • • • • • •• a b c d e • a b c d e• f g • f g • f g • f g• h i c d − a + h e − b + i • h i c d − a + h e − b + i

(155)

RF

(9)12C

(a, . . . , i) =

• • • • • • • • • • • •• a b c • d b a • c b d• e f e • e f e • e f e• g • c − a + g • d − a + g • g • c − a + g • d − a + g• h b h • h b h • h b h• i f c − a + i • d − a + i f i • c − a + i f d − a + i• • • • • • • • • • • •• a b c • d b a • c b d• e f e • e f e • e f e• g • c − a + g • d − a + g • g • c − a + g • d − a + g• h b h • h b h • h b h• i f c − a + i • d − a + i f i • c − a + i f d − a + i

(156)

35

RF

(9)12D

(a, . . . , i) =

• • • • • • • • • • • •• a b c d a • c b a d c• e • f • e • f • e • f• g b g d g • g b g d g• h • c − a + h • h • c − a + h • h • c − a + h• i b f − e + i d i • f − e + i b i d f − e + i• • • • • • • • • • • •• a b c d a • c b a d c• e • f • e • f • e • f• g b g d g • g b g d g• h • c − a + h • h • c − a + h • h • c − a + h• i b f − e + i d i • f − e + i b i d f − e + i

(157)

Thus we have three pairs of cognate families, and the F(9)12A family is self–

cognate.

At least one of the above orbits is permutation equivalent to the orbit con-structed using the Dita’s method:

F(9)12 (α1, . . . , α9) = (158)

[F3]1,1 · F(1)4 (α1) [F3]1,2 · D(α4, α5, α6) · F (1)

4 (α2) [F3]1,3 · D(α7, α8, α9) · F (1)4 (α3)

[F3]2,1 · F(1)4 (α1) [F3]2,2 · D(α4, α5, α6) · F (1)

4 (α2) [F3]2,3 · D(α7, α8, α9) · F (1)4 (α3)

[F3]3,1 · F(1)4 (α1) [F3]3,2 · D(α4, α5, α6) · F (1)

4 (α2) [F3]3,3 · D(α7, α8, α9) · F (1)4 (α3)

where D(α, β, γ) denotes the 4 × 4 diagonal matrix diag(1, eiα, eiβ, eiγ) and

F(1)4 (α) is given by (62).

It passes through F(9)12 (0) = F3⊗F4, which is, according to [59], permutation

equivalent to F12, so F(9)12 is a maximal affine Hadamard family stemming from

a permuted F12.

It also passes through F(9)12 (π/2, π/2, π/2,0), which is dephased, so it is a

permuted F3 ⊗F2 ⊗F2. Thus permuted and inequivalent F12 and F3 ⊗F2 ⊗F2

(see [59]) are connected by the orbit of (158).

Note also that a similar construction using the Dita’s method with the role

of F3 and F(1)4 exchanged yields a 7-dimensional orbit, also passing through a

permuted F12: F4 ⊗F3, which is a suborbit of one of the existing 9-dimensionalmaximal affine Hadamard families stemming from F4 ⊗ F3.

5.12.2 Other 12 × 12 orbits

Other dephased 12×12 orbits can be obtained, using the Dita’s method, from F2

and dephased 6 × 6 complex Hadamard matrices from section 5.6, for example:

36

FD(8)12 (α1, . . . , α8) =

[

[F2]1,1 · F(2)6 (α1, α2) [F2]1,2 · D(α4, . . . , α8) · D(1)

6 (α3)

[F2]2,1 · F(2)6 (α1, α2) [F2]2,2 · D(α4, . . . , α8) · D(1)

6 (α3)

]

(159)

FC(7)12 (α1, . . . , α7) =

[

[F2]1,1 · F(2)6 (α1, α2) [F2]1,2 · D(α3, . . . , α7) · C(0)

6

[F2]2,1 · F(2)6 (α1, α2) [F2]2,2 · D(α3, . . . , α7) · C(0)

6

]

(160)

FS(7)12 (α1, . . . , α7) =

[

[F2]1,1 · F(2)6 (α1, α2) [F2]1,2 · D(α3, . . . , α7) · S(0)

6

[F2]2,1 · F(2)6 (α1, α2) [F2]2,2 · D(α3, . . . , α7) · S(0)

6

]

(161)

DD(7)12 (α1, . . . , α7) =

[

[F2]1,1 · D(1)6 (α1) [F2]1,2 · D(α3, . . . , α7) · D(1)

6 (α2)

[F2]2,1 · D(1)6 (α1) [F2]2,2 · D(α3, . . . , α7) · D(1)

6 (α2)

]

(162)

DC(6)12 (α1, . . . , α6) =

[

[F2]1,1 · D(1)6 (α1) [F2]1,2 · D(α2, . . . , α6) · C(0)

6

[F2]2,1 · D(1)6 (α1) [F2]2,2 · D(α2, . . . , α6) · C(0)

6

]

(163)

DS(6)12 (α1, . . . , α6) =

[

[F2]1,1 · D(1)6 (α1) [F2]1,2 · D(α2, . . . , α6) · S(0)

6

[F2]2,1 · D(1)6 (α1) [F2]2,2 · D(α2, . . . , α6) · S(0)

6

]

(164)

CC(5)12 (α1, . . . , α5) =

[

[F2]1,1 · C(0)6 [F2]1,2 · D(α1, . . . , α5) · C(0)

6

[F2]2,1 · C(0)6 [F2]2,2 · D(α1, . . . , α5) · C(0)

6

]

(165)

CS(5)12 (α1, . . . , α5) =

[

[F2]1,1 · C(0)6 [F2]1,2 · D(α1, . . . , α5) · S(0)

6

[F2]2,1 · C(0)6 [F2]2,2 · D(α1, . . . , α5) · S(0)

6

]

(166)

37

SS(5)12 (α1, . . . , α5) =

[

[F2]1,1 · S(0)6 [F2]1,2 · D(α1, . . . , α5) · S(0)

6

[F2]2,1 · S(0)6 [F2]2,2 · D(α1, . . . , α5) · S(0)

6

]

(167)

where D(β1, . . . , β5) denotes the 6 × 6 diagonal matrix diag(1, eiβ1, . . . , eiβ5).

5.13 N = 13

5.13.1 Orbits stemming from F13

The Fourier matrix F13 is an isolated 13 × 13 complex Hadamard matrix:

F13 = F(0)13 =

1 1 1 1 1 1 1 1 1 1 1 1 11 w w2 w3 w4 w5 w6 w7 w8 w9 w10 w11 w12

1 w2 w4 w6 w8 w10 w12 w w3 w5 w7 w9 w11

1 w3 w6 w9 w12 w2 w5 w8 w11 w w4 w7 w10

1 w4 w8 w12 w3 w7 w11 w2 w6 w10 w w5 w9

1 w5 w10 w2 w7 w12 w4 w9 w w6 w11 w3 w8

1 w6 w12 w5 w11 w4 w10 w3 w9 w2 w8 w w7

1 w7 w w8 w2 w9 w3 w10 w4 w11 w5 w12 w6

1 w8 w3 w11 w6 w w9 w4 w12 w7 w2 w10 w5

1 w9 w5 w w10 w6 w2 w11 w7 w3 w12 w8 w4

1 w10 w7 w4 w w11 w8 w5 w2 w12 w9 w6 w3

1 w11 w9 w7 w5 w3 w w12 w10 w8 w6 w4 w2

1 w12 w11 w10 w9 w8 w7 w6 w5 w4 w3 w2 w

(168)where w = exp(i · 2π/13), so w13 = 1.

5.13.2 Petrescu 13 × 13 orbit

There exists a continuous 2-parameter orbit P(2)13 of 13× 13 complex Hadamard

matrices found by Petrescu [62].

P(2)13 (e, f) = P13 ◦ EXP

(

i · RP

(2)13

(e, f))

(169)

38

where

P13 =

1 1 1 1 1 1 1 1 1 1 1 1 11 −1 t10 −t5 t5 it5 −it5 it15 −it15 t16 t4 t22 t28

1 t10 −1 t5 −t5 −it5 it5 −it15 it15 t16 t4 t22 t28

1 −t5 t5 −1 t10 it15 −it15 it5 −it5 t4 t16 t28 t22

1 t5 −t5 t10 −1 −it15 it15 −it5 it5 t4 t16 t28 t22

1 it5 −it5 it25 −it25 −1 t10 −t5 t5 t22 t28 t4 t16

1 −it5 it5 −it25 it25 t10 −1 t5 −t5 t22 t28 t4 t16

1 it25 −it25 it5 −it5 −t5 t5 −1 t10 t28 t22 t16 t4

1 −it25 it25 −it5 it5 t5 −t5 t10 −1 t28 t22 t16 t4

1 t4 t4 t16 t16 t28 t28 t22 t22 t20 t10 t10 t10

1 t16 t16 t4 t4 t22 t22 t28 t28 t10 t20 t10 t10

1 t28 t28 t22 t22 t16 t16 t4 t4 t10 t10 t20 t10

1 t22 t22 t28 t28 t4 t4 t16 t16 t10 t10 t10 t20

(170)where t = exp(i · 2π/30), so t30 = 1, t15 = −1, and

RP

(2)13

(e, f) = (171)

• • • • • • • • • • • • •

• • • f f e e e + G (f) e + G (f) • • • •

• • • f f e e e + G (f) e + G (f) • • • •

• f f • • e + G (f) e + G (f) e e • • • •

• f f • • e + G (f) e + G (f) e e • • • •

• −e −e −e − G (f) −e − G (f) • • −f −f • • • •

• −e −e −e − G (f) −e − G (f) • • −f −f • • • •

• −e − G (f) −e − G (f) −e −e −f −f • • • • • •

• −e − G (f) −e − G (f) −e −e −f −f • • • • • •

• • • • • • • • • • • • •

• • • • • • • • • • • • •

• • • • • • • • • • • • •

• • • • • • • • • • • • •

where

G (f) = arg

(

−cos(f)

2+ i ·

√2

4

√

7 − cos(2f)

)

− 2π

3(172)

Since the function G(f) is nonlinear, the above family is not an affineHadamard family, but it is not clear whether it could be contained in any affineHadamard family of a larger dimension.

Due to some freedom in the construction of family components the methodof Petrescu allows one to build other similar families of Hadamard matrices.Not knowing whether they are inequivalent we are not going to consider themhere.

5.13.3 ’Cyclic 13–roots’ matrices

There are precisely two ≃ equivalence classes of complex Hadamard matrices,inequivalent to F13, associated with the so–called ’index 2’ cyclic 13–roots. Thisresult is drawn in [21] p.319.

39

The classes are represented by the matrices C(0)13A, C

(0)13B below, their re-

spective dephased forms are denoted by C(0)13A, C

(0)13B. Both matrices have the

circulant structure [U ]i,j = x(i−j mod 13)+1, where

x = [1, c, c, c, c, c, c, c, c, c, c, c, c] for C(0)13A (173)

x = [1, d, d, d, d, d, d, d, d, d, d, d, d] for C(0)13B (174)

and

c =

(

−1 +√

13

12

)

+ i ·(√

130 + 2√

13

12

)

(175)

d =

(

−1 −√

13

12

)

+ i ·(√

130 − 2√

13

12

)

(176)

.Conjugating C

(0)13k, C

(0)13k, k = A, B yields a matrix equivalent to the original

one [21].

The matrices C(0)13A, C

(0)13A and C

(0)13B, C

(0)13B are symmetric. They read

C(0)13A =

1 c c−1 c c c−1 c−1 c−1 c−1 c c c−1 cc 1 c c−1 c c c−1 c−1 c−1 c−1 c c c−1

c−1 c 1 c c−1 c c c−1 c−1 c−1 c−1 c cc c−1 c 1 c c−1 c c c−1 c−1 c−1 c−1 cc c c−1 c 1 c c−1 c c c−1 c−1 c−1 c−1

c−1 c c c−1 c 1 c c−1 c c c−1 c−1 c−1

c−1 c−1 c c c−1 c 1 c c−1 c c c−1 c−1

c−1 c−1 c−1 c c c−1 c 1 c c−1 c c c−1

c−1 c−1 c−1 c−1 c c c−1 c 1 c c−1 c cc c−1 c−1 c−1 c−1 c c c−1 c 1 c c−1 cc c c−1 c−1 c−1 c−1 c c c−1 c 1 c c−1

c−1 c c c−1 c−1 c−1 c−1 c c c−1 c 1 cc c−1 c c c−1 c−1 c−1 c−1 c c c−1 c 1

(177)

40

C(0)13B =

1 d d−1 d d d−1 d−1 d−1 d−1 d d d−1 dd 1 d d−1 d d d−1 d−1 d−1 d−1 d d d−1

d−1 d 1 d d−1 d d d−1 d−1 d−1 d−1 d dd d−1 d 1 d d−1 d d d−1 d−1 d−1 d−1 dd d d−1 d 1 d d−1 d d d−1 d−1 d−1 d−1

d−1 d d d−1 d 1 d d−1 d d d−1 d−1 d−1

d−1 d−1 d d d−1 d 1 d d−1 d d d−1 d−1

d−1 d−1 d−1 d d d−1 d 1 d d−1 d d d−1

d−1 d−1 d−1 d−1 d d d−1 d 1 d d−1 d dd d−1 d−1 d−1 d−1 d d d−1 d 1 d d−1 dd d d−1 d−1 d−1 d−1 d d d−1 d 1 d d−1

d−1 d d d−1 d−1 d−1 d−1 d d d−1 d 1 dd d−1 d d d−1 d−1 d−1 d−1 d d d−1 d 1

(178)

C(0)13A =

1 1 1 1 1 1 1 1 1 1 1 1 11 c−2 c c−3 c−1 c c−1 c−1 c−1 c−3 c−1 c c−3

1 c c2 c c−1 c3 c3 c c c−1 c−1 c3 c1 c−3 c c−2 c−1 c−1 c c c−1 c−3 c−3 c−1 c−1

1 c−1 c−1 c−1 c−2 c c−1 c c c−3 c−3 c−1 c−3

1 c c3 c−1 c c2 c3 c c3 c c−1 c c−1

1 c−1 c3 c c−1 c3 c2 c3 c c c c c−1

1 c−1 c c c c c3 c2 c3 c−1 c c3 c−1

1 c−1 c c−1 c c3 c c3 c2 c c−1 c3 c1 c−3 c−1 c−3 c−3 c c c−1 c c−2 c−1 c−1 c−1

1 c−1 c−1 c−3 c−3 c−1 c c c−1 c−1 c−2 c c−3

1 c c3 c−1 c−1 c c c3 c3 c−1 c c2 c1 c−3 c c−1 c−3 c−1 c−1 c−1 c c−1 c−3 c c−2

(179)

C(0)13B =

1 1 1 1 1 1 1 1 1 1 1 1 11 d−2 d d−3 d−1 d d−1 d−1 d−1 d−3 d−1 d d−3

1 d d2 d d−1 d3 d3 d d d−1 d−1 d3 d1 d−3 d d−2 d−1 d−1 d d d−1 d−3 d−3 d−1 d−1

1 d−1 d−1 d−1 d−2 d d−1 d d d−3 d−3 d−1 d−3

1 d d3 d−1 d d2 d3 d d3 d d−1 d d−1

1 d−1 d3 d d−1 d3 d2 d3 d d d d d−1

1 d−1 d d d d d3 d2 d3 d−1 d d3 d−1

1 d−1 d d−1 d d3 d d3 d2 d d−1 d3 d1 d−3 d−1 d−3 d−3 d d d−1 d d−2 d−1 d−1 d−1

1 d−1 d−1 d−3 d−3 d−1 d d d−1 d−1 d−2 d d−3

1 d d3 d−1 d−1 d d d3 d3 d−1 d d2 d1 d−3 d d−1 d−3 d−1 d−1 d−1 d d−1 d−3 d d−2

(180)

41

5.14 N = 14

5.14.1 Orbits stemming from F14

The only maximal affine Hadamard families stemming from F14 are:

F(6)14 (a, b, c, d, e, f) = F14 ◦ EXP

(

i · RF

(6)14

(a, b, c, d, e, f))

(181)

(

F(6)14 (a, b, c, d, e, f)

)T

= F14 ◦ EXP

(

i ·(

RF

(6)14

(a, b, c, d, e, f))T)

where

F14 =

1 1 1 1 1 1 1 1 1 1 1 1 1 11 w w2 w3 w4 w5 w6 w7 w8 w9 w10 w11 w12 w13

1 w2 w4 w6 w8 w10 w12 1 w2 w4 w6 w8 w10 w12

1 w3 w6 w9 w12 w w4 w7 w10 w13 w2 w5 w8 w11

1 w4 w8 w12 w2 w6 w10 1 w4 w8 w12 w2 w6 w10

1 w5 w10 w w6 w11 w2 w7 w12 w3 w8 w13 w4 w9

1 w6 w12 w4 w10 w2 w8 1 w6 w12 w4 w10 w2 w8

1 w7 1 w7 1 w7 1 w7 1 w7 1 w7 1 w7

1 w8 w2 w10 w4 w12 w6 1 w8 w2 w10 w4 w12 w6

1 w9 w4 w13 w8 w3 w12 w7 w2 w11 w6 w w10 w5

1 w10 w6 w2 w12 w8 w4 1 w10 w6 w2 w12 w8 w4

1 w11 w8 w5 w2 w13 w10 w7 w4 w w12 w9 w6 w3

1 w12 w10 w8 w6 w4 w2 1 w12 w10 w8 w6 w4 w2

1 w13 w12 w11 w10 w9 w8 w7 w6 w5 w4 w3 w2 w

(182)where w = exp(i · 2π/14), so w14 = 1, w7 = −1, and

RF

(6)14

(a, b, c, d, e, f) =

• • • • • • • • • • • • • •• a b c d e f • a b c d e f• • • • • • • • • • • • • •• a b c d e f • a b c d e f• • • • • • • • • • • • • •• a b c d e f • a b c d e f• • • • • • • • • • • • • •• a b c d e f • a b c d e f• • • • • • • • • • • • • •• a b c d e f • a b c d e f• • • • • • • • • • • • • •• a b c d e f • a b c d e f• • • • • • • • • • • • • •• a b c d e f • a b c d e f

(183)

42

F(6)14 and

(

F(6)14

)T

are a pair of cognate families.

At least one of them can be obtained by permuting one of Dita’s construc-tions, either

F(6)14A(α1, . . . , α6) =

[

[F2]1,1 · F7 [F2]1,2 · D(α1, . . . , α6) · F7

[F2]2,1 · F7 [F2]2,2 · D(α1, . . . , α6) · F7

]

(184)

where D(α1, . . . , α6) is the 7 × 7 diagonal matrix diag(1, eiα1 , . . . , eiα6),or

F(6)14B(β1, . . . , β6) (185)

such that its i, j-th 2×2 block is equal to [F7]i,j ·D(α)·F2 , where i, j ∈ {1 . . . 7},

D(α) is the diagonal matrix diag(1, eiα) and α = 0, β1, . . . , β6 for j = 1, 2, . . . , 7respectively.

This is because F(6)14A(0) = F2 ⊗ F7 and F

(6)14B(0) = F7 ⊗ F2 are, according

to [59], permutation equivalent to F14, so both Dita’s orbits are maximal affineHadamard families stemming from permuted F14’s.

5.14.2 Other 14 × 14 orbits

Other dephased 14×14 orbits can be obtained, using the Dita’s method, from F2

and dephased 7 × 7 complex Hadamard matrices from section 5.7, for example:

FP(7)14 (α1, . . . , α7) =

[

[F2]1,1 · F(0)7 [F2]1,2 · D(α2, . . . , α7) · P (1)

7 (α1)

[F2]2,1 · F(0)7 [F2]2,2 · D(α2, . . . , α7) · P (1)

7 (α1)

]

(186)

FC(6)14k(α1, . . . , α6) =

[

[F2]1,1 · F(0)7 [F2]1,2 · D(α1, . . . , α6) · C(0)

7k

[F2]2,1 · F(0)7 [F2]2,2 · D(α1, . . . , α6) · C(0)

7k

]

(187)

PP(8)14 (α1, . . . , α8) =

[

[F2]1,1 · P(1)7 (α1) [F2]1,2 · D(α3, . . . , α8) · P (1)

7 (α2)

[F2]2,1 · P(1)7 (α1) [F2]2,2 · D(α3, . . . , α8) · P (1)

7 (α2)

]

(188)

PC(7)14k(α1, . . . , α7) =

[

[F2]1,1 · P(1)7 (α1) [F2]1,2 · D(α2, . . . , α7) · C(0)

7k

[F2]2,1 · P(1)7 (α1) [F2]2,2 · D(α2, . . . , α7) · C(0)

7k

]

(189)

43

CC(6)14lm(α1, . . . , α6) =

[

[F2]1,1 · C(0)7l [F2]1,2 · D(α1, . . . , α6) · C(0)

7m

[F2]2,1 · C(0)7l [F2]2,2 · D(α1, . . . , α6) · C(0)

7m

]

(190)

where k ∈ {A, B, C, D} and

(l, m) ∈ {(A, A), (A, B), (A, C), (A, D), (B, B), (B, C), (B, D), (C, C), (C, D), (D, D)}

designate 7×7 complex Hadamard matrices associated with cyclic 7–roots, andD(β1, . . . , β6) denotes the 7 × 7 diagonal matrix diag(1, eiβ1, . . . , eiβ6).

5.15 N = 15

5.15.1 Orbits stemming from F15

The only maximal affine Hadamard families stemming from F15 are :

F(8)15 (a, b, c, d, e, f, g, h) = F15 ◦ EXP

(

i · RF

(8)15

(a, b, c, d, e, f, g, h))

(191)

(

F(8)15 (a, b, c, d, e, f, g, h)

)T

= F15 ◦ EXP

(

i ·(

RF

(8)15

(a, b, c, d, e, f, g, h))T)

where

F15 =

1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 w w2 w3 w4 w5 w6 w7 w8 w9 w10 w11 w12 w13 w14

1 w2 w4 w6 w8 w10 w12 w14 w w3 w5 w7 w9 w11 w13

1 w3 w6 w9 w12 1 w3 w6 w9 w12 1 w3 w6 w9 w12

1 w4 w8 w12 w w5 w9 w13 w2 w6 w10 w14 w3 w7 w11

1 w5 w10 1 w5 w10 1 w5 w10 1 w5 w10 1 w5 w10

1 w6 w12 w3 w9 1 w6 w12 w3 w9 1 w6 w12 w3 w9

1 w7 w14 w6 w13 w5 w12 w4 w11 w3 w10 w2 w9 w w8

1 w8 w w9 w2 w10 w3 w11 w4 w12 w5 w13 w6 w14 w7

1 w9 w3 w12 w6 1 w9 w3 w12 w6 1 w9 w3 w12 w6

1 w10 w5 1 w10 w5 1 w10 w5 1 w10 w5 1 w10 w5

1 w11 w7 w3 w14 w10 w6 w2 w13 w9 w5 w w12 w8 w4

1 w12 w9 w6 w3 1 w12 w9 w6 w3 1 w12 w9 w6 w3

1 w13 w11 w9 w7 w5 w3 w w14 w12 w10 w8 w6 w4 w2

1 w14 w13 w12 w11 w10 w9 w8 w7 w6 w5 w4 w3 w2 w

(192)

44

where w = exp(i · 2π/15), so w15 = 1, and

RF

(8)15

(a, b, c, d, e, f, g, h) =

• • • • • • • • • • • • • • •• a b c d • a b c d • a b c d• e f g h • e f g h • e f g h• • • • • • • • • • • • • • •• a b c d • a b c d • a b c d• e f g h • e f g h • e f g h• • • • • • • • • • • • • • •• a b c d • a b c d • a b c d• e f g h • e f g h • e f g h• • • • • • • • • • • • • • •• a b c d • a b c d • a b c d• e f g h • e f g h • e f g h• • • • • • • • • • • • • • •• a b c d • a b c d • a b c d• e f g h • e f g h • e f g h

.

(193)

The families F(8)15 and

(

F(8)15

)T

are cognate.

At least one of them can be obtained by permuting one of Dita’s construc-tions, either

F(8)15A(α1, . . . , α8) =

[F3]1,1 · F5 [F3]1,2 · D(α1, . . . , α4) · F5 [F3]1,3 · D(α5, . . . , α8) · F5

[F3]2,1 · F5 [F3]2,2 · D(α1, . . . , α4) · F5 [F3]2,3 · D(α5, . . . , α8) · F5

[F3]3,1 · F5 [F3]3,2 · D(α1, . . . , α4) · F5 [F3]3,3 · D(α5, . . . , α8) · F5

(194)where D(α1, . . . , α4) is the 5 × 5 diagonal matrix diag(1, eiα1 , . . . , eiα4),

orF

(8)15B(β1, . . . , β8) (195)

such that its i, j-th 3 × 3 block is equal to [F5]i,j · D(α, β) · F3, where i, j ∈{1 . . .5}, D(α, β) is the diagonal matrix diag(1, eiα, eiβ) and

(α, β) = (0, 0), (β1, β2), (β3, β4), (β5, β6), (β7, β8)

for j = 1, 2, . . . , 5 respectively.

This is because F(8)15A(0) = F3 ⊗ F5 and F

(8)15B(0) = F5 ⊗ F3 are, according

to [59], permutation equivalent to F15, so both Dita’s orbits are maximal affineHadamard families stemming from permuted F15’s.

45

5.16 N = 16

5.16.1 Orbits stemming from F16

The only maximal affine Hadamard family stemming from F16 is the 17-parameterorbit:

F(17)16 (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, r) = F16 ◦ EXP(i · R

F(17)16

(a, . . . , r))

(196)where

F16 =

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 w w2 w3 w4 w5 w6 w7 w8 w9 w10 w11 w12 w13 w14 w15

1 w2 w4 w6 w8 w10 w12 w14 1 w2 w4 w6 w8 w10 w12 w14

1 w3 w6 w9 w12 w15 w2 w5 w8 w11 w14 w w4 w7 w10 w13

1 w4 w8 w12 1 w4 w8 w12 1 w4 w8 w12 1 w4 w8 w12

1 w5 w10 w15 w4 w9 w14 w3 w8 w13 w2 w7 w12 w w6 w11

1 w6 w12 w2 w8 w14 w4 w10 1 w6 w12 w2 w8 w14 w4 w10

1 w7 w14 w5 w12 w3 w10 w w8 w15 w6 w13 w4 w11 w2 w9

1 w8 1 w8 1 w8 1 w8 1 w8 1 w8 1 w8 1 w8

1 w9 w2 w11 w4 w13 w6 w15 w8 w w10 w3 w12 w5 w14 w7

1 w10 w4 w14 w8 w2 w12 w6 1 w10 w4 w14 w8 w2 w12 w6

1 w11 w6 w w12 w7 w2 w13 w8 w3 w14 w9 w4 w15 w10 w5

1 w12 w8 w4 1 w12 w8 w4 1 w12 w8 w4 1 w12 w8 w4

1 w13 w10 w7 w4 w w14 w11 w8 w5 w2 w15 w12 w9 w6 w3

1 w14 w12 w10 w8 w6 w4 w2 1 w14 w12 w10 w8 w6 w4 w2

1 w15 w14 w13 w12 w11 w10 w9 w8 w7 w6 w5 w4 w3 w2 w

(197)where w = exp(i·2π/16), so w16 = 1, w8 = −1, w4 = i, and (where typographicpurposes we denoted e − a + k by

(

e − a+k

)

,

so this is not a binomial coefficient)

46

RF

(17)16

(a, . . . , r) = (198)

• • • • • • • • • • • • • • • •

• a b c d e f g • a b c d e f g

• h i j • h i j • h i j • h i j

• k l m d

(

e − a

+k

) (

f − b

+l

) (

g − c

+m

)

• k l m d

(

e − a

+k

) (

f − b

+l

) (

g − c

+m

)

• n • n • n • n • n • n • n • n

• o b

(

c − a

+o

)

d

(

e − a

+o

)

f

(

g − a

+o

)

• o b

(

c − a

+o

)

d

(

e − a

+o

)

f

(

g − a

+o

)

• p i

(

j − h

+p

)

• p i

(

j − h

+p

)

• p i

(

j − h

+p

)

• p i

(

j − h

+p

)

• r l

(

m − k

+r

)

d

(

e − a

+r

) (

f − b

+l

) (

g − c + m

+r − k

)

• r l

(

m − k

+r

)

d

(

e − a

+r

) (

f − b

+l

) (

g − c + m

+r − k

)

• • • • • • • • • • • • • • • •

• a b c d e f g • a b c d e f g

• h i j • h i j • h i j • h i j

• k l m d

(

e − a

+k

) (

f − b

+l

) (

g − c

+m

)

• k l m d

(

e − a

+k

) (

f − b

+l

) (

g − c

+m

)

• n • n • n • n • n • n • n • n

• o b

(

c − a

+o

)

d

(

e − a

+o

)

f

(

g − a

+o

)

• o b

(

c − a

+o

)

d

(

e − a

+o

)

f

(

g − a

+o

)

• p i

(

j − h

+p

)

• p i

(

j − h

+p

)

• p i

(

j − h

+p

)

• p i

(

j − h

+p

)

• r l

(

m − k

+r

)

d

(

e − a

+r

) (

f − b

+l

) (

g − c + m

+r − k

)

• r l

(

m − k

+r

)

d

(

e − a

+r

) (

f − b

+l

) (

g − c + m

+r − k

)

.

The 17–dimensional family F(17)16 of N = 16 complex Hadamard matrices is

self–cognate. Note that this dimensionality coincides with the defect, d(F16) =17, which follows from Eq. (19).

The above orbit is permutation equivalent to the orbit constructed with theDita’s method:

F(17)16 (α1, . . . , α17) =

[

[F2]1,1 · F(5)8 (α1, . . . , α5) [F2]1,2 · D(α11, . . . , α17) · F (5)

8 (α6, . . . , α10)

[F2]2,1 · F(5)8 (α1, . . . , α5) [F2]2,2 · D(α11, . . . , α17) · F (5)

8 (α6, . . . , α10)

]

(199)

where the only maximal affine Hadamard family stemming from F8: F(5)8 such

that F(5)8 (0) = F8, is given by (121), and D(α11, . . . , α17) is the 8 × 8 diagonal

matrix diag(1, eiα11 , . . . , eiα17).

This is because

F16 = F(17)16 (0, (1/16)2π, (2/16)2π, . . . , (7/16)2π) · (200)

[e1, e3, e5, e7, e9, e11, e13, e15, e2, e4, e6, e8, e10, e12, e14, e16]T

where ei denotes the i-th standard basis column vector, so indeed (199) generatesthe only maximal affine Hadamard family stemming from a permuted F16.

The above orbit also passes through F(17)16 (0) = F2 ⊗ F8, as well as through

permuted F2 ⊗ F2 ⊗ F4 and F2 ⊗ F2 ⊗ F2 ⊗ F2, since F(5)8 of (124), or the

47

permutation equivalent F(5)8 of (121), both pass through permuted F2 ⊗F4 and

F2 ⊗F2 ⊗F2. Note that all the tensor products F16, F2 ⊗F8, F2 ⊗F2 ⊗F4 andF2 ⊗ F2 ⊗ F2 ⊗ F2 are inequivalent [59].

6 Closing remarks

Let us summarize our work by proposing a set of dephased representatives GN

of equivalence classes of Hadamard matrices of size N = 2, 3, 4, 5, and by enu-merating the sets from the sum of which one should be able to extract such aset of representatives for N = 6, . . . , 16. The dots indicate that the existenceof other equivalence classes cannot be excluded. For instance, one could lookfor new inequivalent families for composite N using the construction by Dita[22] with permuted some of the component families of Hadamard matrices ofsmaller size.

We tend to believe that the compiled list is minimal in the sense that eachfamily is necessary, since it contains at least some matrices not equivalent to allothers. However, the presented orbits of Hadamard matrices may be (partially)equivalent, and equivalences may hold within families as well as between them.

In the list of complex Hadamard matrices presented below, let {X(d)N } denote

the set of elements of the family X(d)N .

N = 2 G2 = {F (0)2 }.

N = 3 G3 = {F (0)3 }.

N = 4 G4 = {F (1)4 (a); a ∈ [0, π)}.

N = 5 G5 = {F (0)5 }.

N = 6 G6 ⊂

{F (2)6 } ∪

{

(

F(2)6

)T}

∪

{D(1)6 } ∪

{C(0)6 } ∪

48

{S(0)6 } ∪ . . .

N = 7 G7 ⊂

{F (0)7 } ∪

{P (1)7 } ∪

{C(0)7A} ∪

{C(0)7B} ∪

{C(0)7C } ∪

{C(0)7D} ∪ . . .

N = 8 G8 ⊂

{F (5)8 } ∪ . . .

N = 9 G9 ⊂

{F (4)9 } ∪ . . .

N = 10 G10 ⊂

{F (4)10 } ∪

{

(

F(4)10

)T}

∪ . . .

N = 11 G11 ⊂

{F (0)11 } ∪

{C(0)11A} ∪

{C(0)11B} ∪

{N (0)11 } ∪ . . .

N = 12 G12 ⊂

{F (9)12A} ∪

{F (9)12B} ∪

49

{F (9)12C} ∪

{F (9)12D} ∪

{

(

F(9)12B

)T}

∪{

(

F(9)12C

)T}

∪{

(

F(9)12D

)T}

∪

{FD(8)12 } ∪

{FC(7)12 } ∪

{FS(7)12 } ∪

{DD(7)12 } ∪

{DC(6)12 } ∪

{DS(6)12 } ∪

{CC(5)12 } ∪

{CS(5)12 } ∪

{SS(5)12 } ∪ . . .

N = 13 G13 ⊂

{F (0)13 } ∪

{C(0)13A} ∪

{C(0)13B} ∪

{P (2)13 } ∪ . . .

N = 14 G14 ⊂

{F (6)14 } ∪

{

(

F(6)14

)T}

∪

{FP(7)14 } ∪

⋃

k∈{A,B,C,D}{FC(6)14k} ∪

{PP(8)14 } ∪

⋃

k∈{A,B,C,D}{PC(7)14k} ∪

50

⋃

(l,m)∈{(A,A),(A,B),(A,C),(A,D),(B,B),(B,C),(B,D),(C,C),(C,D),(D,D)}{CC(6)14lm} ∪

. . .

N = 15 G15 ⊂

{F (8)15 } ∪

{

(

F(8)15

)T}

∪ . . .

N = 16 G16 ⊂

{F (17)16 } ∪ . . .

Note that the presented list of equivalence classes is complete only for N =2, 3, 4, 5, while for N ≥ 6 the full set of solutions remains unknown. The list ofopen questions could be rather long, but let us mention here some most relevant.

i). Check if there exist other inequivalent complex Hadamard matrices of sizeN = 6.

ii). Find the ranges of parameters of the existing N = 6 families such that allcases included are not equivalent.

iii). Check whether there exists a continuous family of complex Hadamardmatrices for N = 11.

iv). Investigate if all inequivalent real Hadamard matrices of size N = 16, 20belong to continuous families or if some of them are isolated.

v). Find for which N there exist continuous families of complex Hadamard ma-trices which are not affine, and which are not contained in affine Hadamardfamilies of a larger dimension.

vi). Find the dimensionalities of continuous orbits of inequivalent Hadamardmatrices stemming from FN if N is not a power of prime.

Problems analogous to i)–iv) are obviously open for higher dimensions.Thus a lot of work is still required to get a full understanding of the prop-erties of the set of complex Hadamard matrices, even for one–digit dimensions.In spite of this fact we tend to believe that the above collection of matrices willbe useful to tackle different physical problems, in particular these motivated bythe theory of quantum information [43]. Interestingly, the dimension N = 6,the smallest product of two different primes, is the first case for which not allcomplex Hadamard matrices are known, as well as the simplest case for whichthe MUB problem remains open [54].

It is a pleasure to thank B.-G. Englert for his encouraging remarks whichinspired this project. We enjoyed fruitful collaboration on Hadamard matriceswith I. Bengtsson, A. Ericsson, M. Kus and W. S lomczynski. We would like to

51

thank C.D. Godsil, M. Grassl, M. Rotteler, and R. Werner for fruitful discus-sions and P. Dita, M. Matolcsi, R. Nicoara, W. Orrick, and T. Tao for helpfulcorrespondence. K.Z. is grateful for hospitality of the Perimeter Institute forTheoretical Physics in Waterloo, where this work was initiated. We acknowl-edge financial support by Polish Ministry of Science and Information Technologyunder the grant PBZ-MIN-008/P03/2003. We are indebted to W. Bruzda forcreating a web version of the catalogue of complex Hadamard matrices whichmay be updated in future – see http://chaos.if.uj.edu.pl/∼karol/hadamard

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